The 3n+1 Problem

The 3n+1 problem appears to have been first posed by Collatz in 1937.  In this problem, a sequence is generated starting with an initial natural number n.  The rule for generating the next natural number in the sequence is; if n is even, the next natural number in the sequence is n/2, or if n is odd, the next natural number in the sequence is 3n+1.  For example, if the initial value of n is 17, the sequence generated is {17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ...}.  If the natural number 4 is encountered in the sequence, the sequence starts repeating ({4, 2, 1} is generated over and over).  There are three possibilities; (1) the natural number 4 is encountered in the sequence, (2) the sequence starts repeating, but for a different cycle than {4, 2, 1}, or (3) the sequence doesn't repeat (in which case the natural numbers generated in the sequence have to keep getting larger and larger).  The 3n+1 conjecture states that only the first possibility can occur.  Let c be an odd integer greater than or equal to -1 that is not divisible by 3.  The next number in the 3n+c sequence is defined to be 3n+c if n is odd, or n/2 if n is even (the next number in the sequence is always a natural number).  The sequence repeats for {4c, 2c, c} if c>-1 or {2, 1} if c=-1.  In this more general sequence, there are usually cycles other than {4c, 2c, c}.  Although the 3n+1 and 3n-1 sequences have some unique properties, the 3n+c problem is essentially the same for all c values.

The 3n+1 problem is difficult and is not likely to be solvable by an amateur.  Although it might seem that the problem should be classified as "recreational" mathematics, there is a considerable body of mainstream mathematical literature on the subject.  (See Jeffrey C. Lagarias' "The 3x+1 problem:  An annotated bibliography (1963-1999)" at http://arxiv.org/abs/math.NT/0309224 and "The 3x+1 Problem:  An Annotated Bibliography, II (2000-2009)" at http://arxiv.org/abs/math.NT/0608208.  Also, see Lagarias' 1985 article "The 3x+1 problem and its generalizations" at http://www.cecm.sfu.ca/organics/papers/lagarias/index.html.)  This article reviews some of the highlights of the 3n+1 problem literature and presents some original research on the matter (in the form of empirically derived "results").  (For the record, I have a degree in mathematics, but am not a professional mathematician [as is evident from the lack of formal proofs].)  The level of the presentation is elementary and empirically derived propositions are specifically identified when practical.  (Sometimes entire sections are mostly empirically derived.  In these cases, a statement to that effect is made at the beginning of the section.)  Interested readers can verify these propositions for themselves, although fairly sophisticated computer programming skills would be required.

Newcomers to the 3n+1 problem are frequently mystified by how the sequence increases and decreases in a seemingly random fashion, but always appears to arrive at 1.  The fluctuations of the sequence are of little interest once the probability of the situation is taken into account (whether there are cycles other than {4, 2, 1} is far more interesting).  See the section "A probabilistic heuristic" in the Wikipedia article at http://en.wikipedia.org/wiki/Collatz_conjecture for a brief introduction to the subject, the section "A heuristic argument" in Lagarias' 1985 article for more details, and the section "A Random-Walk Argument" in Richard E. Crandall's¹ 1978 article "On the "3x+1" Problem".  The probabilistic argument to be given here avoids the "mixing" assumptions in Lagarias' approach and is amenable to empirical verification.  If the 3n+1 sequence is bounded and there are no cycles other than {4, 2, 1}, then the sequence element 1 must eventually be reached.  The argument to be given is then that the 3n+1 sequence is bounded.

A Probabilistic Argument that the 3n+1 Sequence is Bounded

The probabilistic argument entails a restructuring of the Collatz graph and an alternate definition of the 3n+1 sequence.  The Collatz graph is a tree-like structure that shows how the sequence element 1 can be arrived at starting from different initial n values.  (See the Wikipedia article for the standard depiction of the Collatz graph.  Of course, the graph can only be constructed if the 3n+1 conjecture is assumed to be true.)  When n is even and n-1 is divisible by 3, there is a node in the graph where the previous elements in the sequence are 2n and (n-1)/3.  For example, in the Collatz graph, two limbs ending in 32 and 5 converge to form a limb starting at 16.  In the restructured Collatz graph, 5 is considered to be a continuation of the limb segment starting with 16.  In general, (n-1)/3 is considered to be a continuation of the limb segment at such nodes.  (That is, "odd" paths are taken when tracing back through the nodes.)  The limbs in the restructured Collatz graph are then;

Each limb other than (4, 2, 1) contains exactly one odd element divisible by 3.  Let j denote the odd natural number that is divisible by 3.  The natural numbers to the right of j are not divisible by 3 (since 3 does not divide 3j+1, (3j+1)/2, ...).  The natural numbers to the left of j must be of the form 2ⁱj where i is a natural number (no natural number can attach to this part of the limb since 3 does not divide 2j-1, 4j-1, 8j-1, ...).  (The limbs listed above [starting with the second limb] contain 3, 9, 15, and 21 respectively.  The second limb attaches to the first limb, the third and fifth limbs attach to the second limb, and the fourth limb attaches to the third limb.)  Note that other than the limb {4, 2, 1}, the next-to-last element in a limb is divisible by 8 and that no natural number to the right of the odd natural number divisible by 3 and before the next-to-last natural number is divisible by 8 (this is known as a 1-2 sequence vector since there are either one or two even sequence elements between successive odd sequence elements).

The 3n+c sequence can be defined by the recurrence operation [(3/2)^h(n+c)-c]/2→n, 2^h divides n+c, 2^h+1 does not divide n+c.  Each iteration of the recurrence operation will be referred to as a "jump".  If n is odd and 2² does not divide n+c, then the sequence {n, 3n+c, (3n+c)/2=(3/2)(n+c)-c, ...} is generated where every other element of the sequence up to (3/2)(n+c)-c is odd.  Similarly, if n is odd and 2² divides n+c, 2³ does not divide n+c, then the sequence {n, 3n+c, (3n+c)/2, 3[(3n+c)/2]+c, [3[(3n+c)/2]+c]/2=(3/2)²(n+c)-c, ...} is generated where every other element of the sequence up to (3/2)²(n+c)-c is odd.  In general, if 2^h divides n+c, 2^h+1 does not divide n+c, then the first element in the sequence that is divisible by 4 is the one just before (3/2)^h(n+c)-c.  If [(3/2)^h(n+c)-c]/2 is even, then the first element in the sequence that is divisible by 8 has been found.  Each limb of the restructured Collatz graph other than {4, 2, 1} consists of a series of jumps, the last jump ending in an even natural number (which connects the limb to the rest of the tree).  The question of whether the 3n+1 sequence (or the 3n+c sequence) is bounded then reduces to the question of whether there are any jumps ending in an even natural number.  The distribution of the number of jumps (denoted by i) before an even natural number is reached for the 1000000 sequences starting with 3, 9, 15, ..., 5999997 is;

The probability that i iterations are required is about (1/2)ⁱ.  Even though the probability that the sequence is unbounded is effectively 0, it's unlikely that this part of the 3n+1 conjecture is provable.  (The data indicates that the process which forms 3n+1 sequences [having 1-2 sequence vectors] consisting of an arbitrarily large number of jumps is random.  [In the mathematical literature, such processes are usually said to be "pseudo-random", but in the absence of a rigorous definition of "random", one could argue that saying the process is random is acceptable.]).  The probability that h=1 is about 1/2, the probability that h=2 is about 1/4, the probability that h=3 is about 1/8, etc.  These are the expected probabilities.

In 1972, John H. Conway² showed that a more general function iteration problem similar in form to the 3n+1 problem is computationally undecidable.  This lends some credence to the notion that this part of the 3n+1 problem is unprovable.  However, some progress can be made in this area; in 1976, Riho Terras³ showed that almost all numbers have finite "stopping time".  Let S₀=N and S_i=S_i-1/2 if S_i-1 is even or 3(S_i-1+1)/2 otherwise.  (This is the way the 3n+1 sequence is usually defined in the mathematical literature.  This turns out to be the "natural" way to define the sequence.)  The smallest value of i such that S_i<S₀ is defined to be the stopping time of N. 

Necessary and Sufficient Conditions for Cycles in the 3n+c Sequence to Exist

An element s in the 3n+c sequence can be expressed as (Xa-cZ)/Y where a is a subsequent element in the sequence, X is a natural number and a power of 2, Y is a natural number and a power of 3, and Z is a natural number and a 3-adic number with power-of-two coefficients (see Bohm and Sontacchi's⁴ article and page 41 of Gunther Wirsching's⁵ book [for discussions of the c=1 case] ).  (An example of Z is 340=2²3³+2³3²+2⁵3¹+2⁶3⁰.)  If s=a, then s=cZ/(X-Y).  Other than setting c to X-Y, no method for finding X, Y, and Z where cZ/(X-Y) is an integer is known, so other requirements for cycles will be investigated.

The 3n+c Sequence in the Integer Domain

In the mathematical literature, the cycles {4, 2, 1} for c=1 and {2, 1} for c=-1 are usually said to be "trivial" (but there doesn't appear to be any rationale for this designation).  For a given c value, there appear to be few (if any) non-trivial cycles.  When c is composite, some of the cycles that occur are essentially no different from the cycles that occur for a factor of c (the elements of the cycle are just a common multiple of the elements of the cycle for the factor of c).  (Cycles that are not multiples of other cycles are said to be primitive.  Note that the cycle {4c, 2c, c} for c>1 is not primitive.)  Also, some cycles for a given c value are interrelated; they have the same lengths (number of elements), the same number of odd elements, and approximately the same dynamic range of elements.  Counting these types of cycles as redundant, a histogram of the apparent number of cycles (including the "trivial" cycles) for c values less than or equal to 151 is;

When c=-1, there are two known cycles other than (2, 1); the cycle (20, 10, 5, 14, 7) and the cycle (68, 34, 17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, 272, 136).  A frequently asked question is why there are non-trivial cycles for the 3n-1 sequence, but apparently not for the 3n+1 sequence.  There are many such examples when general negative c values are allowed.  For example, there is at least one primitive cycle for the 3n+7 sequence, but apparently none for the 3n-7 sequence.  However, there appears to be at least one primitive cycle in either the 3n+c sequence or the 3n-c sequence.  This indicates that the distinction between the 3n+c and 3n-c sequences is artificial and that negative n values should be allowed (and that the c values should be required to be positive).   (However, it is sometimes convenient to use the original definition of the 3n+c sequence.)  More evidence that the 3n+1 and 3n-1 sequences (n>0) should be considered to be the same sequence will be presented later.

Properties of Cycles in the 3n+c Sequence

In this section, negative n values are allowed, c is required to be positive, and the element after an odd element i in the 3n+c sequence is defined to be (3i+c)/2.  Let l denote the number of elements in a cycle and m the number of odd elements in the cycle.  An empirical result is;

(1)  A cycle in the 3n+c sequence exists only if c divides 2^l-3^m.

A parity vector gives the order of odd and even elements in a cycle, a 1 for an odd element and a 0 for an even element.  An algorithm for finding all cycles in the 3n+c sequence is to generate all possible parity vectors (distinct under rotation) for given l and m values and to determine which of the corresponding 3n+c cycles where c=|2^l-3^m| are primitive.  This algorithm is based on the following empirical result;

(2)  If l and m are relatively prime,  the distinct parity vectors having a length of l and containing m 1's are in one-to-one correspondence with the primitive cycles in the 3n+c sequence having a length of l and containing m odd elements.  If l and m are not relatively prime, not all parity vectors having a length of l and containing m 1's correspond to cycles in the 3n+c sequence.

For l=1 and m=1, the parity vector is (1) (corresponding to the cycle {-1} for c=1), for l=2 and m=1, the parity vector is (0, 1) (corresponding to the cycle {2, 1} for c=1), and for l=3 and m=2 the parity vector is (1, 1, 0) (corresponding to the cycle {-5, -7, -10} for c=1).  For l=11 and m=7, there are 30 distinct parity vectors, the first few of which are (0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0), (1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0), (1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0), ... (corresponding to cycles for c=139).  The cycle corresponding to the first parity vector is not primitive (when c=139) and corresponds to the remaining known c=1 cycle of {-34, -17, -25, -37, -55, -82, -41, -61, -91, -136, -68}.  Most 3n+1 (n>0) cycle research is concerned with proving that cycles other than {2, 1} don't exist.  In the integer domain, the question is why cycles do exist; there appears to be at least one primitive cycle for every c value.  A more profound question than the original 3n+1 problem is why the values of |2^l-3^m| factor this way.

Another example factorization will be given.  For l=8 and m=4, the distinct parity vectors are (1, 1, 0, 0, 1, 1, 0, 0), (0, 1, 0, 1, 0, 1, 0, 1), (1, 0, 1, 1, 0, 1, 0, 0), (0, 0, 1, 1, 1, 1, 0, 0), (0, 1, 1, 0, 1, 1, 0, 0), (1, 0, 0, 1, 1, 1, 0, 0), (0, 1, 0, 1, 1, 1, 0, 0), (1, 0, 1, 0, 1, 1, 0, 0), (1, 1, 0, 1, 0, 1, 0, 0), and (0, 1, 1, 1, 0, 1, 0, 0) (corresponding to cycles for c=175).  The first two parity vectors do not correspond to primitive cycles for any c value, the third parity vector corresponds to a primitive cycle for c=25, the fourth and fifth parity vectors correspond to primitive cycles for c=35, and the remaining parity vectors correspond to primitive cycles for c=175.  Although 5 and 7 also divide 175, there are no primitive cycles for these c values when l=8 and m=4 (the parity vectors have been used up by cycles for the larger c values).  Note that 1 divides 2^l-3^m for every l and m value, but this c value behaves like any other c value; there appear to be only finitely many primitive cycles for a given c value. 

The first parity vector in the above example consists of the duplicated (twice) sub-vector (1, 1, 0, 0) and the second parity vector consists of the duplicated (four times) sub-vector (0, 1) and the number of times the sub-vectors are duplicated divides the greatest common divisor of l and m (4).  When l=9 and m=6, the parity vector (0, 1, 1, 0, 1, 1, 0, 1, 1) does not correspond to a primitive cycle and the sub-vector (0, 1, 1) is duplicated three times.  When l=12 and m=8, the parity vectors (0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1), (0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1), and (0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1) do not correspond to primitive cycles.  In the first parity vector, the sub-vector (0, 0, 1, 1, 1, 1) is duplicated twice, in the second parity vector, the sub-vector (0, 1, 0, 1, 1, 1) is duplicated twice, and in the third parity vector, the sub-vector (0, 1, 1) is duplicated four times.  For l=12 and m=1, 2, 3, ..., 11, the numbers of distinct parity vectors are 1, 6, 19, 43, 66, 80, 66, 43, 19, 6, and 1 respectively and the numbers of primitive 3n+c cycles are 1, 5, 18, 40, 66, 75, 66, 40, 18, 5, and 1 respectively.  An empirical result is;

(3)  Parity vectors containing duplicated sub-vectors do not correspond to unduplicated 3n+c cycles.

A cycle with no even elements can occur only if (3/2)ⁱ(n+c)-c=n where n is odd, i<h, and 2^h is the largest power of 2 that divides n+c ((3/2)ⁱ(n+c)-c is an incomplete jump).  The only primitive cycle with no even elements is the cycle {-1} for c=1 (where c=-n).  An m-cycle is a hypothetical cycle in the 3n+1 sequence (n>0) having m local minima.  (In this article, such cycles in the 3n+c sequence will be referred to as being M-cycles.)  In 1977, R. P. Steiner⁶ proved that 1-cycles can't exist.  A jump in the 3n+c sequence ending in the initial odd element is analogous to a 1-cycle (there would be exactly one even element in the cycle).  Empirical results are;

(4)  A cycle in the 3n+c sequence having only one even element can occur only if c=|2^l-3^l-1|.

(5)  A cycle in the 3n+c sequence having only one odd element can occur only if c=|2^l-3|.

Cycles in the 3n+c sequence where the number of even elements equals the number of odd elements and c properly divides |2^l-3^m| frequently occur.  For a given number of elements in a cycle, a cycle for a small factor of |2^l-3^m| is more likely to occur when the number of even and odd elements are approximately equal.  For example, the values of |2^l-3^m| (factored into primes) for l=18 and m=1, 2, 3, ..., 17 are 262141=11∙23831, 262135=5∙103∙509 (a primitive cycle occurs for c=52427=103∙509), 262117=61∙4297, 262063=503∙521, 261901=23∙59∙193 (primitive cycles occur for c=193, 1357=23∙59, 4439=23∙193, and 11387=59∙193), 261415=5∙7²∙11∙97 (primitive cycles occur for c=97, 539=7²∙11, 679=7∙97, 1067=11∙97, 2695=5∙7²∙11, 4753=7²∙97, 7469=7∙11∙97, and 37345=5∙7∙11∙97), 259957=47∙5531 (primitive cycles occur for c=47 and 5531), 255583= 431∙593 (primitive cycles occur for c=431 and 593), 242461=37∙6553 (primitive cycles occur for c=6553), 203095=5∙151∙269 (primitive cycles occur for c=151, 269, 755=5∙151, and 40619=151∙269), 84997=11∙7727 (a primitive cycle occurs for c=7727), 269297=7∙17∙31∙73 (primitive cycles occur for c=119=7∙17, 527=17∙31, 1241=17∙73, 2263=31∙73, 3689=7∙17∙31, 15841=7∙31∙73, and 38471=17∙31∙73), 1332179=61∙21839 (a primitive cycle occurs for c=21839), 4520825=5²∙67∙2699 (primitive cycles occur for c=67475=5²∙2699, 180833=67∙2699, and 904165=5∙67∙2699), 14086763=179∙78697, 42784577=11∙23∙263∙643, and 128878019=563∙228913 respectively.  (As expected, primitive cycles also occur for these l and m values when c=|2^l-3^m|.)  For a large number of elements in a cycle, the smallest factor of |2^l-3^m| for which a primitive cycle can occur is "large" (how to further quantify this statement isn't evident).  Considering this, it's surprising that there appears to be a primitive cycle for every c value.  Another empirical result is;

(6)  The sign of the elements of a cycle in the 3n+c sequence cannot change.  The signs of the elements in interrelated 3n+c cycles are the same (the sign is determined by 2^l-3^m).

When the element following an odd element i in the 3n+c sequence is defined to be (3i+c)/2 instead of 3i+c, 1-2 sequence vectors become 0-1 sequence vectors.  Note that 2m must be greater than l for there to be any cycles having 0-1 sequence vectors.  For example, when l=11 and m=7, the parity vectors of cycles having 0-1 sequence vectors are (1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1), (1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0), (1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1), (1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1), and (1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1) (corresponding to primitive cycles for c=139).  The 3n+139 cycle corresponding to the second parity vector is {-4151, -6157, -9166, -4583, -6805, -10138, -5069, -7534, -3767, -5581, -8302} and twice the largest element in the cycle (-3767) is greater than the smallest odd element (-6805).   In the other interrelated 3n+139 cycles having 0-1 sequence vectors, twice the largest element in a cycle is not greater than the smallest odd element in the cycle.  Let min denote the element of a cycle in the 3n+c sequence having the smallest absolute value and let max denote the odd element having the largest absolute value.  More empirical results are;

(7)  If c=|2^l-3^m| and 2m>l, there is at least one primitive cycle having a 0-1 sequence vector and there are usually primitive cycles not having 0-1 sequence vectors.  If c=|2^l-3^m|, 2m>l, and the greatest common divisor of l and m is 1, there is at least one cycle among the interrelated primitive cycles having 0-1 sequence vectors where 2|min|>|max| (there is no cycle among the interrelated primitive cycles not having 0-1 sequence vectors where 2|min|>|max|).  If c=|2^l-3^m|, 2m<l, and gcd(l, m)=1, there is at least one cycle among the interrelated primitive cycles where 2|min|>|max|.

(8)  2|min|>|max| for a primitive 3n+c cycle where gcd(l, m)=1 only if c=|2^l-3^m|.

Usually, if c=|2^l-3^m|, 2m>l, and gcd(l, m)=1, there is exactly one cycle among the interrelated primitive cycles having 0-1 sequence vectors where 2|min|>|max|.  When c=1675, l=9, and m=7, there are two primitive cycles where 2|min|>|max|.  For one of these cycles, min=-2219, max=-4429, and 2|min| is barely greater than |max| (min=-2363 and max=-3997 for the other cycle).  Usually, if c=|2^l-3^m|, 2m>l, and gcd(l, m)≠1, there is at least one cycle among the interrelated primitive cycles having 0-1 sequence vectors where 2|min|>|max|.  When c=9823, l=14, and m=8, there doesn't appear to be a primitive cycle where 2|min|>|max|.  If c=|2^l-3^m|, 2m<l, and gcd(l, m)=1, there appears to be exactly one cycle among the interrelated primitive cycles where 2|min|>|max|.  If c=|2^l-3^m|, 2m<l, and gcd(l, m)≠1, there doesn't appear to be a primitive cycle where 2|min|>|max|.  Usually, if a primitive 3n+c cycle where c properly divides 2^l-3^m exists and gcd(l, m)≠1, then 2|min| is not greater than |max|.  When c=145, l=12, and m=8, min=-617, max=-1231, and 2|min| is barely greater than |max|.  Also, when c=493, l=12, and m=8, min=-1829, max=-3635, and 2|min| is barely greater than |max|.

In 1997, Halbeisen and Hungerbühler⁷ proved optimal estimates for the length of a Collatz cycle in terms of its minimum using the formula M_l,m=∑(]jm/l[ - ](j-1)m/l[)2^j-13^m-]jm/l[ where the summation is from j=1 to l and the reversed brackets denote the ceiling function (the minimum is M_l,m/(2^l-3^m)).  As will be proved, twice the minimum element in such a cycle (one having the parity vector ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l) must be larger than the maximum odd element.  (Many details of the proof are omitted so that only a sketch of the proof remains.  The proof just mimics Halbeisen and Hungerbühler's proof.)  Another empirical result is;

(9)  If 2|min|>|max| for a primitive 3n+c cycle, gcd(l, m)=1, and there are no other interrelated primitive cycles where 2|min|>|max|, then the cycle has the parity vector ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l.  If there are interrelated primitive cycles where 2|min|>|max| and gcd(l, m)=1, then only one of the cycles has the parity vector ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l.

For a Collatz cycle, M_l,m/(2^l-3^m) is larger than the minimum element in other cycles having a length of l and containing m odd elements.  Let t_j=[jm/l] - [(j-1)m/l], j=1, 2, 3, ..., l, where the brackets denote the floor function and let u₁=t_l and u_j+1=t_j,  j=1, 2, 3, ..., (l-1) (that is, u is a right-rotation of t by one position).  The analogous formula for the maximum odd element in the cycle is N_l,m=∑u_j2^j-13^m-∑u(k) where the summations are from j=1 to l and k=1 to j (due to typographical difficulties, u(k) is used to denote u_k).  Let r denote gcd(l, m).  The parity vector given in the first formula (that is, ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l) consists of r identical sub-vectors.  Similarly, the parity vector t consists of r identical sub-vectors and each of these sub-vectors is the same as the corresponding sub-vector given by the first formula except for the first and last elements.  First suppose that l and m are relatively prime.  Twice the minimum element minus 3^m-1/(2^l-3^m) then equals the maximum odd element in the cycle.  (When the parity vector from the first formula is right-rotated by one position [corresponding to a multiplication by 2], it matches the parity vector from the second formula except for the first two elements of each sub-vector.  The first mismatch corresponds to a loss of 3^m-1/(2^l-3^m) and the second mismatch corresponds to a gain of 2∙3^m-1/(2^l-3^m).)  In general, twice the minimum element minus ∑2^i(l/r)3^m-1-i(m/r)/(2^l-3^m) where the summation is from i=0 to r-1 equals the maximum odd element in the cycle.  N_l,m/(2^l-3^m) is smaller than the maximum odd element in other Collatz cycles having a length of l and containing m odd elements.  Another empirical result is;

(10)  If c=|2^l-3^m|,  gcd(l, m)=1, and the elements of the interrelated 3n+c cycles are positive,  M_l,m is greater than or equal to the minimum elements in the interrelated 3n+c cycles (not necessarily primitive) and N_l,m is less than or equal to the maximum odd elements in the interrelated 3n+c cycles.  Analogous results apply if the elements of the cycles are negative.

By Proposition (7), if c=|2^l-3^m| and gcd(l, m)=1, there exists at least one primitive 3n+c cycle where 2|min|>|max| and by Proposition (9) the parity vector of one of these cycles is given by ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l.  For this cycle, |min|=M_l,m and |max|=N_l,m.  For example, when c=139, l=11, and m=7, M_l,m=3767 and N_l,m=6805.  By Proposition (10), 3767/139 (approximately equal to 27) must be greater than or equal to the minimum absolute value of elements in the c=1 cycle (17) and 6805/139 (approximately equal to 49) must be less than or equal to the maximum absolute value of odd elements in the c=1 cycle (91). 

In his 1978 article, Crandall proved a lower bound of 17985 for the number of odd elements in a 3n+1 cycle (n>0) using the minimum in a hypothetical cycle and the continued-fraction convergents of log(3)/log(2).  Recent 3n+1 cycle research frequently involves use of the continued-fraction convergents of log(3)/log(2).  For some purposes, this is too restrictive; valuable information is thrown away.  This is the rationale for defining generalized continued-fraction convergents (allowing less accurate approximations).  Let (c, d), (e, f), and (g, h) denote successive continued-fraction convergents of a real number and let j denote the ceiling of h/f.  (ei, fi) and (ei+c, fi+d), i=1, 2, 3, ..., j, can be considered to be generalized continued-fraction convergents of the real number.  (Sometimes it is convenient to consider just (ei+c, fi+d), i=1, 2, 3, ..., j, to be the generalized continued-fraction convergents of the real number; ei+c and fi+d are then relatively prime.)  The generalized continued-fraction convergents of log(3)/log(2) are (2, 1), (3, 2), (4, 2), (5, 3), (6, 4), (8, 5), (9, 6), (11, 7), (16, 10), (19, 12), (24, 15), (27, 17), (38, 24), (46, 29), (57, 36), (65, 41), (76, 48), (84, 53), ....  An empirical result is;

(11)  The absolute values of 2^l-3^m increase monotonically for (l, m) values that are generalized continued-fraction convergents of log(3)/log(2) (excluding (2, 1), (4, 2), (6, 4), and (9, 6)).

The significance of this is that for a given c value, there can be at most one solution of c=|2^l-3^m| where (l, m) are generalized continued-fraction convergents of log(3)/log(2) (excluding (2, 1), (4, 2), (6, 4), and (9, 6)).

A successor of a, a>0, in the 3n+c sequence is greater than a if 3^ma>2^la-cA (that is, 3^m/2^l>1-cA/(2^la)) where the values of l, m, and the integer A depend on the parity vector between a and the successor of a.  For a sufficiently large a, the successor of a is larger than a if 3^m/2^l>1.  For a sufficiently large initial sequence value a, the following algorithm generates the parity vector (denoted by p) of a 3n+c sequence (with positive elements) where twice the minimum element in the sequence is larger than the maximum odd element in the sequence.  Set x to 1, p₁ to 1, and i to 2.  Then repeat the following operations.  Set p_i to 1, x to (3/2)x, and increment i.  If x>1, set p_i to 0, x to (1/2)x, and increment i.  The resulting parity vector is 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, ....  The portion of the parity vector up to the Mth 0 gives an M-cycle.  (The first 1 in the parity vector is taken to follow the 0 so that a cycle is formed.)  The resulting (l, m) values (where l is the number of elements in the cycle and m is the number of odd elements) for the M-cycles are (3, 2), (5, 3), (8, 5), (11, 7), (13, 8), (16, 10), (19, 12), (21, 13), (24, 15), (27, 17), (30, 19), (32, 20), (35, 22), (38, 24), (40, 25), (43, 27), (46, 29), (49, 31), (51, 32), (54, 34), (57, 36), (59, 37), (62, 39), (65, 41), (68, 43), (70, 44), (73, 46), (76, 48), (78, 49), (81, 51), (84, 53), ....  Empirical results are;

(12)  These (l, m) values include the generalized continued-fraction convergents of log(3)/log(2) (except for (2, 1), (4, 2), (6, 4), and (9, 6)).  Also, 2m>l.

(13)  For some rotation of the portion of p up to the Mth 0, an M-cycle has the parity vector ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l.

(14)  The difference in l values of a pair of successive M-cycles is 2 or 3 and the difference in m values is 1 or 2.

If the difference in l values is 2, 3M_l,m+2^l=M_l+2,m+1, otherwise 9M_l,m+3∙2^l-1+2^l+1=M_l+3,m+2 (due to the new elements in the parity vector).  If the difference in l values is 2, (M_l,m+2^l/3)/|(4/3)2^l-3^m| equals M_l+2,m+1/|2^l+2-3^m+1|, otherwise (M_l,m+2^l-1/3+2^l+1/9)/|(8/9)2^l-3^m| equals M_l+3,m+2/|2^l+3-3^m+2|.  Note that the value of M_l,m/|2^l-3^m| increases if the difference in l values is 3 and 2^l-3^m and 2^l+3-3^m+2 have the same sign. 

(15)  There are either 3 or 4 successive M-cycles where the difference in l values is 3 (2^l-3^m is negative for the third or fourth M-cycle and positive otherwise).

(16)  When there are 4 successive M-cycles where the difference in l values is 3, the values of  M_l,m/|2^l-3^m| increase up until the third M-cycle and then decrease.  The absolute values of 2^l-3^m usually increase, but may decrease for the third M-cycle.  When there are 3 successive M-cycles where the difference in l values is 3, the values of M_l,m/|2^l-3^m| usually increase, but may decrease for the third M-cycle.  The absolute values of 2^l-3^m usually increase, but may decrease for the third M-cycle (but only if the value of M_l,m/|2^l-3^m| increases).  After the third or fourth successive M-cycle where the difference in l values is 3, the value of M_l,m/|2^l-3^m| decreases and the absolute value of 2^l-3^m increases (the difference in l values is 2). 

The absolute values of 2^l-3^m where (l, m) are multiplies of a continued-fraction convergent (as usually defined) of log(3)/log(2) increase monotonically since the values of M_l,m/|2^l-3^m| are the same.  Truncated M_l,m/(2^l-3^m) values corresponding to the generalized continued-fraction convergents of log(3)/log(2) (excluding the multiples of convergents) are;

Let K denote the number of odd elements in a 3n+1 cycle (n>0) and L the number of even elements.  In 2004, Simons⁸ proved that if a nontrivial 2-cycle exists, then (K+L, K) must be a convergent in the continued-fraction expansion of log₂ 3.  Let δ denote log(3)/log(2).  Using an empirically derived lower bound for the minimum element in an m-cycle, Simons and de Weger⁹ proved that δK<K+L<1.000001δK.  As shown above, 3n+c cycles with large elements and having a constrained dynamic range behave similarly, that is, the (K+L, K) values include the generalized continued-fraction convergents of log(3)/log(2).  This will be discussed in more detail in the next section.

Characterization of Cycles in the 3n+c Sequence

Attachment points in a cycle are defined to be even integers i such that i-c is divisible by 3 (and (i-c)/3 is not already in the cycle).  An attachment point i will be said to be primary if 4i is not an attachment point.  If i is a primary attachment point and 4 divides i, then i/4 will be said to be the secondary attachment point, etc.  In the cycle for c=1, -68 is the only attachment point.  (There are no attachment points in the other known 3n+1 cycles.)  In the following, 3n+c sequences are extended backwards from attachment points.  "Odd" paths are taken, that is, if i is an even integer in the extended sequence and 3 divides i-c, then the path (i-c)/3 (and not 2i) is taken.  Presumably, an odd integer divisible by 3 (denoted by t) will be reached.  The t value of a primary attachment point will be said to be the "proxy" of the first odd element in the cycle after the preceding primary attachment point (in the case where there is only one primary attachment point, the preceding primary attachment point is the primary attachment point).  (Note that the sequence vector of the cycle elements starting with the first odd element after the preceding primary attachment point and ending with the primary attachment point is a 0-1 sequence vector; this is the motivation for taking "odd" paths when extending sequences backwards from attachment points.  For the c=1 cycle, t equals -21 and is the proxy of the cycle element -17.)  As will be shown, the proxy concept allows a simplified description of  3n+c cycles with attachment points.  The extended sequences have many properties and, unlike Bohm and Sontacchi's formula, are amenable to Diophantine analysis.  Cycles with at least one attachment point will be considered in this section and the next three sections (cycles with no attachment points will be discussed at the end of the article).  An empirical result is;

(17)  There exists at least one primitive 3n+c cycle having an attachment point for every c value.

Let u denote the first odd element in a 3n+c cycle after a primary attachment point.  In this article, the average of the |u| values and the absolute values of their proxies is used to characterize a 3n+c cycle.  Let a denote the number of primary attachment points in a cycle.  When a>0, the average of the |u| values and the absolute values of their proxies will be taken to be the maximum likelihood estimator for the parameter λ of an exponential probability distribution (f(x; λ)=λe^-λx, x≥0).  The function g(x)=λe^-λx where positive and negative x values are allowed will be used to compute the domain of the absolute values of the u values and the absolute values of their proxies.  The domain of the absolute values of the u values and the absolute values of their proxies is determined by -log(|u₁|/λ)/λ, -log(|t₁|/λ)/λ, -log(|u₂|/λ)/λ, -log(|t₂|/λ)/λ, -log(|u₃|/λ)/λ, -log(|t₃|/λ)/λ, ..., -log(|u_a|/λ)/λ, -log(|t_a|/λ)/λ.  For example, the (u, t) values for a cycle for c=121 are (19, -9), (395, 51), (65, 39), (119, 183), (335, 147), (281, 27), (101, -237), (53, -63), (35, 21), and (23, -159).  A plot of the |u| and |t| values versus their domain is;

(The x values have been scaled up by a factor of 10⁴ and all but two data points are shown.  No shape-preserving interpolation of the data points is done.)  Note that all the u and t values in a cycle are distinct (although two t values can have the same absolute value).  The domain of the absolute values of the u values and the absolute values of their proxies appears to be a small interval about zero, the largest x value usually being larger than the absolute value of the smallest x value (so that the maximum |u| or |t| value times the minimum |u| or |t| value is usually less than λ²).  An empirical result (based on the 22982 cycles for the c values less than or equal to 9997) is;

(18)  The maximum |u| or |t| value times the minimum |u| or |t| value divided by λ² is less than log(3)/log(2).

The maximum |u| or |t| value times the minimum |u| or |t| value divided by λ² is the largest (1.580708) for a cycle for c=3013 having four primary attachment points and a (K+L, K) value of (38, 24) [a generalized continued-fraction convergent of log(3)/log(2)].  log(3)/log(2) equals 1.5849625.)  The rationale for including the absolute values of the proxies in the average is that they appear to have the same exponential curve as the |u| values.  Of course, |u| values are local minima and their average is related to the minimum in a cycle (although, unlike an m-cycle, an odd element preceded by exactly two even elements is not considered to be a local minimum).

Belaga¹⁰ proved that if 2^l-3^k>0 where l is the length of and k is the number of odd elements in a 3x+d cycle (x>0), then 1 ≤ n < d/(2^l/k-3) where n is the smallest odd element in the cycle.  Let min denote the element in a 3n+c cycle having the smallest absolute value.  An empirical result is;

(19)  The minimum |u| or |t| value in a 3n+c cycle is almost always less than c/|2^(K+L)/K-3|.

For the 22982 cycles for the c values less than or equal to 9997, only 11 exceptions occur and in every instance, a=1.  When c=29, 1021, and 2045, u=1, K=1, and c=2^(K+L)/K-3.  The remaining exceptions occur for cycles for c=605, 973, and 7645 where L=8 and K=14, a cycle for c=2761 where L=15 and K=25, a cycle for c=3013 where L=14 and K=24, and three cycles for c=7153 where L=7 and K=12 (in every instance |u| and |t| are larger than |min|).  Except for the cycle for c=2761, 2^(K+L)/K-3 is negative.  Let X₀ denote the smallest |u| or |t| value in a cycle.  An empirical result is;

(20)  |1-c/X₀| is usually greater than |2^(K+L)/K-3|.

For the 1555 cycles for the c values less than or equal to 997, 36 exceptions occur and in 26 of these instances, a=1 (2^(K+L)/K-3 is usually positive).  An empirical result (based on the 22982 cycles for the c values less than or equal to 9997) is;

(21)  If (K+L, K) is a generalized continued-fraction convergent of log(3)/log(2), |1-c/X₀| is greater than |2^(K+L)/K-3|.

The chain equation for an m-cycle is given by 3^k(i)a_i-1=2^k(i+1)+l(i)a_i+1-2^l(i), i=0, 1, 2, ..., m-1, where k(i) is the number of successive odd numbers leading up to a local maximum (due to typographical difficulties, k(i) is used to denote k_i), l(i) is the number of successive even numbers going down to the next local minimum (due to typographical difficulties, l(i) is used to denote l_i), and a_i are natural numbers (see Simons and de Weger's article).  The chain equation is the starting point of m-cycle theory and is used to derive Simons and de Weger's Lemma 4;

        0 < Λ < ∑1/x_i

where x_i is a local minimum, the summation is from i=1 to m, and Λ denotes (K+L) log(2)-K log(3).  The above characterization of a 3n+c cycle is related to this lemma.  For most 3n+c cycles, c times the sum of the reciprocals of |u| and |t| is greater than the absolute value of Λ.  When (K+L, K) is equal to (or approximately equal to) a generalized continued-fraction convergent of log(3)/log(2), c times the sum of reciprocals is sometimes smaller than the absolute value of Λ.  An empirical result (based on the 22982 cycles with attachment points for the c values less than or equal to 9997) is;

(22)  3c∑(1/|u_i| + 1/|t_i|) where the summation is from i=1 to a is almost always greater than |Λ|.

This proposition sometimes fails for cycles with one or two primary attachment points.  The proposition fails for a one attachment-point cycle for c=2761 where (K+L, K)=(40, 25), a one attachment-point cycle for c=3013 where (K+L, K)=(38, 24), a one attachment-point cycle for c=4513 where (K+L, K)=(44, 32), a one primary attachment-point cycle for c=7645 where (K+L, K)=(44, 28), a one primary attachment-point cycle for c=8059 where (K+L, K)=(38, 24), a one attachment-point cycle for c=8059 where (K+L, K)=(38, 24), and a two primary attachment-point cycle for c=8059 where (K+L, K)=(38, 24).

Simons and de Weger's Corollary 5 (of Lemma 4) is;

        0 < Λ < m/x_min ≤ m/X₀

where x_min is the global minimum of an m-cycle and X₀ is an empirically derived lower bound for x_min.  (As will be shown, the chain equation is not entirely applicable to the above characterization of a 3n+c cycle, so even if Proposition (22) is accepted, a proposition analogous to Corollary 5 can't be derived without some knowledge of the relative sizes of the |u| and |t| values.)  Simons and de Weger show that Λ is exponentially small in terms of its coefficients and use a result from transcendence theory due to Rhin¹¹ to derive Lemma 12;

        Λ > e^{-13.3(0.46057+log K)}.

Using Corollary 5 gives Simons and de Weger's Corollary 13 (where the global minimum of an m-cycle is estimated in terms of K);

        x_min < me^{13.3(0.46057+log K)}

In the following, all cycles are assumed to have been found for c values up to certain limits.  For the c values less than 1000, at least one primitive cycle where a>1 occurs for every c value except 1, 7, and 37.  The largest domain of the absolute values of the u values and the absolute values of their proxies occurs for a cycle for c=5; the domain is (-2.027326e-001, 3.465736e-001).  The domains of the cycles for most c values are one, two, or three orders of magnitude smaller.  For the c values less than 1000, the smallest domain occurs for a cycle for c=467; the domain is (-1.051072e-007, 1.944498e-007).  Small domains also occur for cycles for c=311 and c=343 (and other c values).  The smallest domains for cycles for larger c values appear to become increasingly smaller.  For example, for c=4501, nine primitive cycles where a>1 occur and the domains are as large as (-3.956808e-004, 1.654824e-003) and as small as (-7.278549e-008, 2.117179e-007).

When the domain of the absolute values of the u values and the absolute values of their proxies is small, the (K+L, K) value of the cycle is likely to be a generalized continued-fraction convergent of log(3)/log(2).  (Even for the c=1 cycle, the (K+L, K) value [(11, 7)] is a generalized continued-fraction convergent of log(3)/log(2).)  For example, when c=467, the (K+L, K) value is (84, 53), a continued-fraction convergent of log(3)/log(2).  When c=311, the (K+L, K) value is (19, 12), a continued-fraction convergent of log(3)/log(2).  When c=343, the (K+L, K) value is (149, 94), a generalized continued-fraction convergent of log(3)/log(2).  When the domain is small, (K+L, K) values that are small multiples (usually 2) of generalized continued-fraction convergents of log(3)/log(2) occur.  For example, when c=4501, the (K+L, K) value is (92, 58), twice a generalized continued-fraction convergent of log(3)/log(2).  When the domain is small, (K+L, K) values that are approximately equal to generalized continued-fraction convergents of log(3)/log(2) occur.  For example, for a cycle for c=407, the (K+L, K) value is (44, 28), almost equal to a generalized continued-fraction convergent ((46, 29)) of log(3)/log(2).  However, cycles occur for some c values where the domain appears to be small but there is no apparent relationship between the (K+L, K) values and the generalized continued-fraction convergents of log(3)/log(2).  This occurs for a cycle for c=1813 where the domain is (-7.915834e-007, 3.418000e-006) and the (K+L, K) value is (228, 132) and for a cycle for c=2009 where the domain is (-4.947314e-006, 1.590942e-005) and the (K+L, K) value is (175, 98).  The (K+L, K) values are fairly large multiples of approximations of generalized continued-fraction convergents of log(3)/log(2).  The domains are no longer "small" for all of the cycles of these large c values.

Extended Sequences of 3n+c Cycles

In this section, the element after an odd element i is defined to be 3i+c.  The shortest possible jump from an odd element in the 3n+c sequence to another odd element will be referred to as a "hop" (when the element after an odd element i is defined to be 3i+c, there are two even elements between the initial odd element and the destination of the jump).  An example of a "multiple-jump" attachment point (for c=11) is -21→-13→-7→-1, 8.  The attachment point is several jumps away from the odd integer divisible by 3, that is, -21 jumps to -13, -13 jumps to -7, -7 jumps to -1, and 8 immediately follows -1.  The jumps in multiple-jump attachment points usually consist of a jump followed by one or more hops or one or more hops followed by a jump.  On average, there are 1.2 jumps and 2.0 hops in a multiple-jump (based on the 3767 multiple-jump attachment points for the c values less than or equal to 1999).  One-jump attachment points (such as -21→-23, -68 [for c=1]) have some special properties.

An example of a "jumped-over" attachment point (for c=13) is 51→358.  The destination of the jump is not the attachment point associated with the odd integer divisible by 3; the attachment point (262 in this case) has been "jumped over".  Empirical results are;

(23)  The destination of a jumped-over attachment point is another attachment point if the destination is even.

(24)  The sign in an extended sequence can change only once.

If the sign in the extended sequence of a jumped-over attachment point changes, it usually does so at the attachment point.  For example, -63→74 for a cycle for c=95 where the attachment point 26 is jumped over.  The extended sequence up to 26 is {-63, -94, -47, -46, -23, 26}.  Sometimes there are several jumps before the attachment point is jumped over.  For example, -249→-287→-173→-89→118 for a cycle for c=169 where the attachment point 22 is jumped over.  The sequence up to 22 (starting with -89) is {-89, -98, -49, 22}.  Jumped-over attachment points become rare when the domain of the absolute values of the u values and the absolute values of their proxies becomes small.  As will be shown (by way of empirical evidence), the largest possible l_i value (in the chain equation) in a 3n+1 cycle (n>0) without a jumped-over attachment point is 5.  (Of course, determining whether jumped-over attachment points cease to exist for a sufficiently small domain would be difficult.)

An example of a "no-jump" attachment point (for c=11) is -3, 2.  The odd integer divisible by 3 is immediately before the attachment point.  Empirical results are;

(25)  In a cycle having an attachment point, there is at least one no-jump or one-jump attachment point.

(26)  In primary, secondary, tertiary, etc., attachment points, the types of attachment points that can occur have a specific order; a no-jump attachment point (not necessarily the primary attachment point) is followed by a jumped-over attachment point, the jumped-over attachment point is followed by a multiple-jump (or one-jump) attachment point, the multiple-jump (or one-jump) attachment point is followed by a no-jump attachment point, etc.

(27)  Except when the last attachment point in primary, secondary, tertiary, etc., attachment points is a jumped-over attachment point, the destination of a jumped-over attachment point is the next attachment point.

(28)  A jumped-over attachment point cannot be a primary attachment point.

In the chain equation, the element after an odd element i in the 3n+1 sequence (n>0)  is defined to be (3i+1)/2.  Counting the even element before a primary attachment point and the even elements up to the tertiary attachment point gives 6 even elements.  If there are primary, secondary, and tertiary attachment points in a cycle, one of them must be a jumped-over attachment point (by Proposition (26)).

Solving for a one-jump attachment point followed by a no-jump attachment point (in primary, secondary, tertiary, etc., attachment points) gives the Diophantine equation (3/2)^h=(8t₂+3c)/(t₁+c) where 2^h is the largest power of 2 that divides t₁+c.  Solving for a no-jump attachment point followed by a jumped-over attachment point gives (3/2)ⁱ=(3/8)(t₂+3c)/(t₃+c) (usually, 2ⁱ is not the largest power of 2 that divides t₃+c).  Solving for a jumped-over attachment point followed by a one-jump attachment point gives (3/2)^i-j=3(t₄+c)/(t₃+c), etc.

Hardy and Littlewood¹² proved that the sequence {frac(xⁿ)} where frac(x) is the fractional part of x is equidistributed for almost all real numbers x>1 (the exceptional set has Lesbeque measure zero).  (A sequence of real numbers {x_n} is equidistributed on an interval [a, b] if the probability of finding x_n in any subinterval is proportional to the subinterval length.  The points of an equidistributed sequence form a dense set on the interval [a, b].)  The properties of {frac(3/2)ⁿ} have been extensively studied.  Since {frac(3/2)ⁿ} appears to be equidistributed (and thus dense), the equation (3/2)^h=(8t₂+3c)/(t₁+c) (or one of the other equations) is likely to have many solutions (as can be easily verified).

The exponents of 3/2 in the above equations are small.  For example, for c=41, the attachment points for primary, secondary, tertiary, and quatenary attachment points for a cycle are 3→29, 128 (a one-jump attachment point), -3, 32 (a no-jump attachment point), -21→2 (an attachment point where 8 is jumped over [the sequence up to 8 is {-21, -22, -11, 8}]), and -87→-55→-31→-13, 2 (a multiple-jump attachment point).  The last attachment point is not a one-jump attachment point, but -31 can be substituted for t₄ in the equation above.  t₁=3, t₂=-3, and t₃=-21 and the values of h, i, and j are 2, 2, and 1 respectively.  For c=137, there are primary, secondary, tertiary, and quatenary attachment points for a cycle (starting with a one-jump attachment point) and the successive exponents of 3/2 are 4, 2, and 1.  For c=107, there are primary, secondary, tertiary, etc., attachment points for a cycle (starting with a no-jump attachment point) and the successive exponents of 3/2 are 2, 1, 1, and 3.  For c=4159, there are primary, secondary, tertiary, etc., attachment points for a cycle (starting with a one-jump attachment point) and the successive exponents of 3/2 are 1, 3, 2, 2, 2, and 1.  For c=4519, there are primary, secondary, tertiary, etc. attachment points for a cycle (starting with a one-jump attachment point) and the successive exponents of 3/2 are 2, 2, 1, 1, 2, and 1.  When there are more than three attachment points in primary, secondary, tertiary, etc., attachments points, the attachment points are frequently powers of 2 (for c≤151, the only exception occurs for the above cycle for c=137).

Let locmin denote the odd element between two successive primary attachment points having the smallest absolute value.  Usually, for a one-jump attachment point, locmin equals u or is a few successive hops away from u.  For the 4234 primary one-jump attachment points for the c values less than or equal to 997, the numbers of attachment points where 0, 1, 2, 3, 4, 5, and 6 hops are required to reach locmin are 3677, 358, 102, 19, 5, 1, and 0 respectively.  For 72 of the attachment points, locmin can't be reached by successive hops away from u.  Let locmax denote the odd element between two successive primary attachment points having the largest absolute value.  Let i denote the last odd element before the second primary attachment point.  If locmax≠i, the attachment point (that is, the second primary attachment point) is usually a no-jump attachment point, but can occasionally be a one-jump attachment point (but not a jumped-over or multiple-jump attachment point [an empirical result]).  For the c values less than or equal to 997, there are 1622 primary attachment points where locmax≠i and 1489 of these are no-jump attachment points and 133 are one-jump attachment points.  Another empirical result is;

(29)  A primary attachment point is a multiple-jump attachment point only if the attachment point is one jump away from the first odd cycle element after the preceding primary attachment point.

A consequence of the above proposition (and Proposition (28)) is that a u value jumps (in one jump) to the next primary attachment point, or if its proxy isn't the odd integer immediately before the next primary attachment point, the proxy jumps (in one jump) to the odd integer immediately before the next primary attachment point, etc.  This is the aforementioned simplified structure of a cycle.  Let 2^j be the largest power of 2 that divides the difference between a u value and its proxy.  Empirical results are;

(30)  For a one-jump attachment point, the expected value of j+1 is the number of odd elements in the jump (counting the destination of the jump).

(31)  For a multiple-jump attachment point that is a primary attachment point, j is small (the numbers of j values equal to 1, 2, 3, 4, 5, and 6 for the c values less than or equal to 997 are 555, 447, 82, 9, 1, and 1 respectively). 

There is usually a one-jump attachment point in a cycle where j or j+1 equals the number of odd elements in the jump.  (For the c=1 cycle, -21≡-17(mod 2²) and there are 3 odd elements in the jump.)  For example, when c=3013, there are 93 primitive cycles (with a total of 338 attachment points) having an (K+L, K) value of (38, 24) (a generalized continued-fraction convergent of log(3)/log(2)).  For 72 of the 202 primary one-jump attachment points, j+1 equals the number of odd elements in the jump and for 48 of the primary one-jump attachment points, j equals the number of odd elements in the jump.  A histogram of the differences between the number of odd elements in the jump and j is;

Note that the modulo operations (u≡t(mod 2^j)) introduce variables corresponding to the a_i values in the chain equation (in the chain equation, x_i≡-1(mod 2^k(i)) where x_i is a local minimum).  Similarly, the odd integers before the attachment points in no-jump attachments points correspond to a_i values so that there are the same number of variables as in the chain equation.  From this perspective, the c=1 cycle is a 1-cycle.  [The variables of the chain equation are k₀=4, l₀=1, a₀=-1, k₁=3, l₁=3, and a₁=-5.])  However, these equations don't have the same form as the chain equation (except for multiple-jump attachment points where the usual "path" is taken) since, for a one-jump attachment point, the attachment point equals 3i+c where i is the destination of the jump.  Similarly, these equations don't have the same form as the chain equation for no-jump attachment points.  Simons and de Weger show that all x_i are about the same size by "chaining" them.  Their Lemma 6 is;

        For all i = 0,1,..., m-1 we have x_i+1 < b^δx_i^δ

where b = (1 + X₀^-1)/2^1/δ and X₀ is a lower bound of x_min.  For a one-jump attachment point, the next u value equals {[(3/2)^f+1(t+c)-(c/2)]/2}/2^g where 2^f is the largest power of 2 that divides t+c.  This expression is less than (1/2)(3/2)^f+1(t+c)=(1/2)(2^f+1)^δ-1(t+c).  If u and t are positive, t=2^ja+u where a≥1, so the next u value is less than (1/2)[2^f+1-j(2^ja+u+c)]^δ-1(t+c)=(2^f+1-j)^δ-1(1/2)(t+c)^δ.  So if u and t are positive and f+1 (the number of odd elements in the jump [counting the destination of the jump]) is less than or equal to j, a result analogous to Lemma 6 can be derived (where b=(1+cX₀^-1)/2^1/δ and X₀ is a lower bound for t_min).  For a no-jump attachment point, the next u value equals [(3/2)(t+c)-c]/2^g.  This expression is less than (1/2)(3/2)(t+c), so if u and t are positive, a result analogous to Lemma 6 can be derived.  For a multiple-jump attachment point consisting of a jump followed by a hop, the odd element immediately before the attachment point is [(3/2)^f+1(t+c)-(c/2)]/2².  The next u value then equals {[(3/2)^f+2(t+c)+(5/4)c]/2²}/2^g.  This expression is not necessarily less than (1/2)(3/2)^f+2(t+c).  For a multiple-jump attachment point consisting of a jump followed by two hops, the next u value equals {[(3/2)^f+3(t+c)+(47/8)c]/2³}/2^g.

Simons and de Weger's Lemma 7 (derived from Lemma 6 and Corollary 5) is;

        0 < Λ < mc_m2^{-((δ -1)/ζ)K}

where c_m= 2^{(m/δ)(δ -1)/ζ}b^{δ/(δ-1)-m/ζ} and ζ denotes δ^m-1.  (The symbol ζ is used due to typographical difficulties.)  Simons and de Weger's Lemma 14 (their main result) is;

   Let x=K₁(m) be the largest solution of e^{-13.3(0.46057+log x)}=mc_m2^{-((δ -1)/ζ)x}.  Then K<K₁(m).

The expected number of odd elements in the jump (or jumps in the case of a multiple-jump attachment point) from t to the odd integer before a primary attachment point (including the destination of the jump or t in the case of a no-jump attachment point) is the number of even elements after the even element immediately before the primary attachment point (including the even element) plus the number of jumps (none for a no-jump attachment point, 1 for a one-jump attachment point, and at least 2 for a multiple-jump attachment point) minus the number of hops between u and the primary attachment point.  For example, for the c=1 cycle, there are 3 odds elements in the jump from -21 to -23, the even elements are -136, -68, and -34, and the hop between -17 and -68 is {-55, -164, -82, -41} (3=3+1-1).  A histogram of the number of odd elements in the jump (or jumps) from t to the odd integer before the primary attachment point minus the number of jumps plus the number of hops between u and the primary attachment point minus the number of even elements after the even element before the primary attachment point for the 22843 primary attachment points for the c values less than or equal to 1999 is;

Other than being discrete-valued, this appears to be a normal probability distribution with a mean of -0.005121917 and a standard deviation of 2.378364.  For c values less than or equal to 49, the mean is 0.5319 with a 95% confidence interval of (0.1571, 0.9067) and the standard deviation is 1.8299 with a 95% confidence interval of (1.6005, 2.1367).  A normal probability plot of the data is;



For c values less than or equal to 199, the mean is 0.3458 with a 95% confidence interval of (0.1694, 0.5221) and the standard deviation is 2.4152 with a 95% confidence interval of (2.2968, 2.5466).  A normal probability plot of the data is;



A histogram of the sum of the number of odd elements in the jumps from t values to the odd elements before the primary attachment points minus the sum of the number of jumps plus the number of hops in the cycle minus the sum of the number of even elements after the even element before the primary attachment points for the 3212 cycles less than or equal to 1999 is;

The above distribution has a mean of -0.0364259 and a standard deviation of 6.149562.  The number of odd elements in the jumps from t to the odd element before a primary attachment point and the number of even elements after the even element before the primary attachment point are "complementary" variables.  A histogram of the sum of the number of odd elements in the jumps from the t values to odd elements before the primary attachment points minus the sum of the number of even elements after even elements before the primary attachment points for the 3212 cycles less than or equal to 1999 is;

The above distribution has a mean of -0.4623288 and a standard deviation of 6.645549.  Let c_max denote an upper bound of c values.  A graph of c_max versus log(c_max) minus the standard deviation of this distribution for all c values less than c_max for c_max values of 600, 1000, 1500, 2000, 2500, 3000, 3300, 3500, 3800, 4000, 4300, 4500, 4800, 5000, 5300, 5500, 5800, 6000, 6300, 6500, 6800, 7000, 7200, 7400, 7600, 7800, 8000, 8200, 8400, 8600, 8800, 9000, 9200, 9400, 9600, 9800, and 10000 is;

The number of jumps from t to the odd element before a primary attachment point and the number of hops between the corresponding u value and the primary attachment point are complementary variables.  A histogram of the number of hops in the cycle minus the sum of the number of jumps from t values to the odd elements before the primary attachment points for the 3212 cycles less than or equal to 1999 is;

The above distribution has a mean of 0.4259029 and a standard deviation of 5.237414.  A graph of c_max versus log(c_max) minus the standard deviation of this distribution for all c values less than c_max for c_max values of 600, 1000, 1500, 2000, 2500, 3000, 3300, 3500, 3800, 4000, 4300, 4500, 4800, 5000, 5300, 5500, 5800, 6000, 6300, 6500, 6800, 7000, 7200, 7400, 7600, 7800, 8000, 8200, 8400, 8600, 8800, 9000, 9200, 9400, 9600, 9800, and 10000 is;

Let n denote the number of jumps in the multiple-jump of a primary multiple-jump attachment point.  Also, let n equal 1 for a primary one-jump attachment point and let n equal 0 for a primary no-jump attachment point.  Let h denote the number of hops between u and the primary attachment point, let f denote the number of odd elements in the jump (or jumps) from t to the odd element immediately before the attachment point (let f equal 0 for a no-jump attachment point), and let g denote the number of even elements after the even element before the primary attachment point.  As previously shown, {[(3/2)^f+1(t+c)]/2ⁿ}/2^g is less than the next u value for no-jump and one-jump attachment points and greater than the next u value for multiple-jump attachment points.  Assuming these deficiencies and excesses cancel out, ∏{[(3/2)^f(i)+1(t_i+c)]/2ⁿ⁽ⁱ⁾}/2^g(i) is approximately equal to ∏u_i where the product is from i=1 to a.  [2^{∑(f(i)+1)-∑j(i)}]^δ-1/2^∑g(i) is usually less than 1, so if the u and t values are positive, ∏(t_i+c )^δ/2ⁿ⁽ⁱ⁾ should be greater than ∏u_i.  (Of the 3212 cycles for the c values less than or equal to 1999, [2^{∑(f(i)+1)-∑j(i)}]^δ-1/2^∑g(i) is greater than 1 in only 37 instances and its largest value is 17.31905.)  In proving Lemma 6, Simons and de Weger show that (1/2)(x_i+1)^δ>x_i+1.  Similarly, if the u values are positive, (1/2)(u_i+c)^δ is greater than the next local minimum (possibly the destination of a hop).  An empirical result is;

(32)  The u values can be chained together, that is, |u_i+c|^δ is almost always greater than |u_i+1|.  The corresponding  t values can be chained together, that is, 2ⁿ⁽ⁱ⁾|t_i+c|^δ is almost always greater than |t_i-1|.  Also, 2ⁿ⁽ⁱ⁾|t_i+c|^δ is almost always greater than |u_i+1|.  (The u and t values are indexed circularly.)

Note that there are no restrictions on the signs of the u and t values in the above proposition.  Of the 22843 primary attachment points of the c values less than or equal to 1999, the chain of u values is broken only 23 times (in every instance, the next primary attachment point doesn't have an associated secondary attachment point).  Of the 22843 primary attachment points of the c values less than or equal to 1999, the chain of proxies is broken only 47 times.  The t value of a primary multiple-jump attachment point where the multiple-jump consists of two jumps, two hops, or a jump and a hop can be chained (without scaling by a factor of 2²) with the t value of the preceding attachment point (not necessarily primary).  (If |t+c|=2^h, then the absolute value of the preceding t value equals 3^h [an empirical result] so that |t+c|^δ equals the absolute value of the preceding t value.)  Chaining one of these t values to the t value of the preceding primary attachment point is then apt to fail (even after scaling by a factor of 2²) when the preceding primary attachment point has associated secondary, tertiary, quatenary, etc. attachment points.  This accounts for 38 out of the 47 instances where the chain of proxies is broken.  In 3 other instances, the t value of a primary one-jump attachment point can't be chained to the t value of the preceding primary attachment point.  This occurs when the preceding primary attachment point has associated secondary and tertiary (and quatenary) attachment points.  Of the 22843 primary attachment points of the c values less than or equal to 1999, 2ⁿ⁽ⁱ⁾|t_i+c|^δ is less than |u_i+1| in 99 instances and 96 of these instances occur when the multiple jump of a primary multiple-jump attachment point consists of two jumps, two hops, or a jump and a hop.  The corresponding definition of b in Lemma 6 is b=|1+cX₀^-1| where X₀ is multiple-valued, that is, if u or t is positive, then u or t is greater than X₀, or if u or t is negative, then u or t is less than -X₀.  (Note that b=1+cX₀^-1 or |1-cX₀^-1|.)  m-cycles then become a-cycles.  c_a is then defined so that |1+cX₀^-1|^ab^{(δ/(δ-1))(ζ/(δ-1)-a)}=c_a^ζ/(δ-1), that is, c_a=b^{δ/(δ-1)-a/ζ} where ζ denotes δ^a-1.  ∑(1/|u_i|+1/|t_i|) where the summation is from i=1 to a is less than a/|u|_min+a/|t|_min, so the corresponding version of Lemma 7 is;

(33)  |Λ|<3cac_a2^{-((δ-1)/ζ)K}

Of course, empirically determining a lower bound for the proxies presents some difficulties.  For the c=121 cycle, L=42, K=46, a=10, the largest negative proxy value is -9, the smallest positive proxy value is 21, and the smallest u value is 19.  Λ equals 10.46078661.  Assuming this is the only c=121 10-cycle in existence, X₀ can be set to 9.  b then equals 14.44444444 or 12.44444444.  c_a then equals 1059.577577 or 718.2799973.  3cac_a2^{-((δ-1)/ζ)K} then equals 3186070 or 2159814.  This test (where X₀ is set to the smallest |u| or |t| value in the cycle) fails for 93 of the 18302 cycles with attachment points for the c values less than or equal to 7999.  In every case, the (K+L, K) value of the cycle is approximately equal to a generalized continued-fraction convergent of log(3)/log(2) and the a value is small.  Usually, the cycle is a 1-cycle where the inequality simplifies to |Λ|<3cb2^-K.  Setting the X₀ value to 17 for the c=1 cycle gives 3cb2^-K equals 0.0248161 or 0.0220588 (Λ=-0.06566703).  When c=23, the test fails for a 3-cycle where (K+L, K)=(43, 26), when c=295, the test fails for a 2-cycle where (K+L, K)=(30, 18), when c=679, the test fails for a 5-cycle where (K+L, K)=(60, 36), when c=2369, the test fails for a 2-cycle where (K+L, K)=(24, 14), when c=2381, the test fails for a 5-cycle where (K+L, K)=(50, 30), when c=2647, the test fails for a 4-cycle where (K+L, K)=(55, 32), when c=2999, the test fails for a 2-cycle where (K+L, K)=(47, 28), when c=3221, the test fails for a 2-cycle where (K+L, K)=(35, 21), when c=3623, the test fails for a 3-cycle where (K+L, K)=(38, 23), when c=4741, the test fails for a 3-cycle where (K+L, K)=(32, 19), when c=4741, the test fails for a 2-cycle where (K+L, K)=(32, 19), when c=6511, the test fails for a 2-cycle where (K+L, K)=(37, 22), and when c=6661, the test fails for a 5-cycle where (K+L, K)=(56, 33).  (None of these (K+L, K) values exactly equals a generalized continued-fraction convergent of log(3)/log(2).)  In every instance, the test passes when b is set to 1+cX₀^-1 and fails when b is set to |1-cX₀^-1| (the respective values of 3cac_a2^{-((δ-1)/ζ)K} in the latter case are 0.79, 0.22, 0.24, 0.76, 0.88, 1.64, 1.42, 0.13, 0.11, 0.85, 0.0036, 0.20, and 0.38 and the respective values of |Λ| are 1.24, 1.02, 2.04, 1.25, 1.70, 2.97, 1.82, 1.19, 1.07, 1.31, 1.31, 1.48, and 2.56).   In every instance, the smallest |u| or |t| value in the cycle is approximately equal to c (the smallest |u| or |t| values are 33, 281, 669, 2399, 2349, 2589, 3985, 3187, 3609, 4779, 4743, 6441, and 6621 respectively).  When X₀ is set to infinity, the comparison fails for 66 of the 18302 cycles with attachment points for the c values less than or equal to 7999 and in every instance the cycle is a 1-cycle.  The inequality then appears to be valid for all cycles except 1-cycles.

In the following, the objective is to show that the expected distribution of the differences between the number of odd elements in a primary one-jump attachment point and the corresponding j value for a fixed c value becomes too narrow for there to be any cycles when the domain of the absolute values of the u values and the absolute values of their proxies becomes sufficiently small.  A histogram of the differences between the number of odd elements in primary one-jump attachment points and the j values for the c=467 cycles where the (K+L, K) values are (84, 53) is;

A histogram of the differences between the number of odd elements in primary one-jump attachments points and the j values for the c=311 cycles where the (K+L, K) values are (19, 12) is;

As expected, the shoulders on the peak for the c=311 cycles are broader than those on the peak for the c=467 cycles (taking into account the different numbers of primary one-jump attachment points).  When j is greater than the number of odd elements in the jump, locmin is usually not equal to u.  An empirical result is;

(34)  When j equals the number of odd elements in a primary one-jump attachment point, locmin has to equal u.

When locmin doesn't equal u, the difference between the number of odd elements in the jump and the j value ranges from -11 to 10 for the c values less than or equal to 997.  The corresponding numbers of attachment points are 1, 0, 0, 1, 2, 8, 15, 7, 30, 69, 113, 0, 25, 118, 89, 45, 18, 8, 2, 2, 3, and 1.  The corresponding numbers of attachment points where locmin can't be reached by successive hops away from u are 0, 0, 0, 0, 0, 1, 0, 0, 4, 5, 6, 0, 25, 12, 11, 5, 2, 0, 0, 0, 1, and 0.  Note that these attachment points account for all of the attachment points where j+1 equals the number of odd elements in the jump and that most of these attachment points occur when j is less than the number of odd elements in the jump.  A histogram of the number of odd elements in the jump minus the j value for the primary one-jump attachment points where locmin doesn't equal u and locmin can't be reached by successive hops away from u for the c values less than or equal to 3997 is;

This distribution has a mean of 1.056657 and a standard deviation of 2.295541.  A histogram of the number of odd elements in the jump minus the j value for the primary one-jump attachment points where locmin doesn't equal u but locmin can be reached by successive hops away from u for the c values less than or equal to 3997 is;

This distribution has a mean of .6050493 and a standard deviation of 2.969974.  When j is greater than the number of odd elements in the jump, locmax is usually not equal to the last odd element before the attachment point.  An empirical result is;

(35)  When j equals the number of odd elements in a primary one-jump attachment point, locmax has to equal the last odd element before the attachment point.

When locmax doesn't equal the last odd element before the attachment point, the difference between the number of odd elements in the jump and the j value ranges from -6 to 2  for the c values less than or equal to 997.  The corresponding numbers of attachment points are 1, 5, 0, 5, 10, 24, 0, 87, and 1.  A general explanation for why cycles can't occur when the distribution of the differences between the number of odd elements in a primary one-jump attachment point and the corresponding j value becomes too narrow is that they would become too uniform (locmin would have to equal u and locmax would have to equal the last odd element before the attachment point).  For primary multiple-jump attachment points, locmin equals u and locmax equals the last odd element before the attachment point.  For a primary no-jump attachment point, the probability that j equals 1 is about 1/2, the probability that j equals 2 is about 1/4, the probability that j equals 3 is about 1/8, etc.  For the c values less than or equal to 997, the numbers of primary no-jump attachment points where j equals 1, 2, 3, ..., 12 are 1265, 627, 302, 152, 83, 37, 18, 18, 3, 1, 2, and 0 respectively.  For the c=467 cycles where (K+L, K) equals (84, 53), the difference between the number of odd elements in the jump (1) and the j value ranges from -4 to 0.  The corresponding numbers of attachment points are 1, 4, 7, 6, and 14.

A histogram of j minus the number of jumps from t to the odd integer before the primary attachment point plus the number of hops between u and the primary attachment point minus the number of even elements after the even element before the primary attachment point for the 22843 primary attachment points for the c values less than or equal to 1999 is;

j and the number of even elements after the even element before a primary attachment point are complementary variables.  A histogram of j minus the number of even elements after the even element before the primary attachment point for the 22843 primary attachments points for the c values less than or equal to 1999 is;

For c values less than or equal to 499, this distribution has a mean of -0.5521875 and a standard deviation of 2.070596 (there are 3200 primary attachment points).  For c=467 (and (K+L, K)=(84, 53)), the distribution has a mean of 0.1891892 and a standard deviation of 2.180548 (there are 111 primary attachment points).  Since there are no jumped-over attachment points, the differences between the j values and the numbers of even elements after the even element before the primary attachment point are skewed to the positive side.  (The differences range from -3 to 9 and the histogram values are 1, 15, 39, 21, 15, 8, 4, 1, 3, 1, 0, 2, and 1 respectively.)  Apparently, there is a limit to how skewed the distribution can be for a given c value.  For c=107, there is apparently only one cycle and the (K+L, K) value is (106, 53).  Since the number of even and odd elements in the cycle are the same, the differences between the j values and the numbers of even elements after the even element before the primary attachment point are skewed to the negative side.  The distribution has a mean of -1.4 and a standard deviation of 3.098387 (there are 10 primary attachment points).  (The differences range from -9 to 2 and the histogram values are 1, 0, 0, 0, 0, 0, 1, 2, 2, 1, 2, and 1 respectively.)  In cycles having the same number of even and odd elements and where the number of odd elements equals the denominator of a generalized continued-fraction convergent of log(3)/log(2), j minus the number of even elements after the even element before the primary attachment point is no more negative (approximately) than the difference for cycles having the (K+L, K) value of a generalized continued-fraction convergent of log(3)/log(2) is positive (an empirical result). 

A histogram of the sum of the j values minus the sum of the number of even elements after the even element before the primary attachment point for the 3212 cycles with attachment points for the c values less than or equal to 1999 is;

This distribution has a mean of -5.275841 and a standard deviation of 8.975339.  The difference between the sum of the j values and the sum of the numbers of even elements after the even element before the primary attachment point is likely to be the most negative (and should be even more negative for larger c values) in cycles having more even elements than odd elements.  The largest difference appears to be bounded or to very slowly increase as c increases.

A histogram of j minus the number of even elements after the even element before the primary attachment point where the (K+L, K) value of the cycle equals a generalized continued-fraction convergent of log(3)/log(2) for the c values less than or equal to 9997 is;

This distribution has a mean of 0.06927298 and a standard deviation of 2.013401.  A histogram of the sum of the j values minus the sum of the numbers of even elements after the even element before the primary attachment point where the (K+L, K) value of the cycle equals a generalized continued-fraction convergent of log(3)/log(2) for the c values less than or equal to 9997 is;

This distribution has a mean of 0.1379405 and a standard deviation of 2.86086.  For a given cycle, the sum of the j values minus the sum of the numbers of even elements after the even element before the primary attachment point appears to be less than or equal to 17.  In cycles having the (K+L, K) value of a generalized continued-fraction convergent of log(3)/log(2), the differences between the j values and the numbers of even elements after the even element before the primary attachment point are mostly uncorrelated (the dynamic range of the domain of the above histograms is about the same).  The maximum a value that a cycle can have for a given upper bound of c values can then be estimated from the mean of the distribution of j minus the number of even elements after the even element before the primary attachment point for this upper bound.  For example, for c values less than or equal to 2999, a should be less than 180.09 (17/0.09439528).  For c values less than or equal to 2497, a should be less than 155.91 (17/0.1090343).  For c values less than or equal to 1499, a should be less than 139.63 (17/0.1217532).  For c values less than or equal to 599, a should be less than 122.64 (17/0.1386139).  Denote these estimated a values by a´.  For the c upper bounds of 2997, 2497, 1499, and 599, the a´/log(c) values are 22.5, 19.9, 19.1, and 19.2 respectively.  An empirical result is;

(36)  For a 3n+c cycle having attachment points and a (K+L, K) value equal to a generalized continued-fraction convergent of log(3)/log(2), 3∙log(c)+1≥a.

This proposition is based on the 3661 cycles having attachment points and (K+L, K) values equal to generalized continued-fraction convergents of log(3)/log(2) for c less than or equal to 9997.  (When c=343, 3·log(c)=17.5 and a=15 for a cycle where (K+L, K)=(149, 94).  Other than the cycle for c=1, this is the smallest difference found.)

A histogram of j minus the number of even elements after the even element before the primary attachment point where the number of even and odd elements in the cycle are the same and the K value equals the denominator of a generalized continued-fraction convergent of log(3)/log(2) for the c values less than or equal to 7999 is;

This distribution has a mean of -0.7588988 and a standard deviation of 1.924729.  A histogram of the sum of the j values minus the sum of the numbers of even elements after the even element before the primary attachment point where the number of even and odd elements in the cycle are the same and the K value equals the denominator of a generalized continued-fraction convergent of log(3)/log(2) for the c values less than or equal to 7999 is;

This distribution has a mean of -3.790436 and a standard deviation of 5.723903.  For a given cycle for c less than or equal to 3997, the sum of the j values minus the sum of the numbers of even elements after the even element before the primary attachment point is greater than or equal to -19.  For c upper bounds of 3997, 3499, 2999, 2497, 1999, 1499, 997, 499, and 125, a should be less than 28 (-19/-0.6705119), 28 (-19/-0.6836965), 29 (-19/-0.6533575), 29 (-19/-0.6471936), 28 (-19/-0.6791604), 26 (-19/-0.7259259), 23 (-19/-0.8162162), 22 (-19/-0.8757396), and 16 (-19/-1.222222) respectively.  (For the c=107 cycle, a=10.)  The largest K value (equal to the denominator of a generalized continued-fraction convergent of log(3)/log(2) and where the number of even and odd elements are the same) found for the c values less than or equal to 3997 is 53.  For the above c upper bounds of 3997, 3499, 2999, 2497, 1999, 1499, 997, 499, and 125, the a´/log(c) values are 3.4, 3.4, 3.6, 3.8, 3.7, 3.6, 3.4, 3.6, and 3.2 respectively (these ratios are part of the rationale for using the factor of 3 in Proposition (36)).  For a cycle for c=4291 having 306 even elements and 306 odd elements, the sum of the j values minus the sum of the numbers of even elements after the even element before the primary attachment point is -63.  Also, for a cycle for c=4381 having 48 even elements and 48 odd elements, the difference is -22, for a cycle for c=4907 having 200 even elements and 200 odd elements, the difference is -28, and for a cycle for c=7427 having 212 even elements and 212 odd elements, the difference is -64.  An empirical result is;

(37)  For a 3n+c cycle having attachment points, the same number of even and odd elements, and a K value equal to the denominator of a generalized continued-fraction convergent of log(3)/log(2), 3∙log(c) is almost always greater than a.

This proposition is based on the 711 cycles having attachment points, the same number of even and odd elements, and K values equal to the denominator of generalized continued-fraction convergents of log(3)/log(2) for c values less than or equal to 7999.  The proposition fails for a cycle for c=4291 having 306 even elements and 306 odd elements, for a cycle for c=4907 having 200 even elements and 200 odd elements, for a cycle for c=7427 having 212 even elements and 212 odd elements, and for a cycle for c=7711 having 200 even elements and 200 odd elements ((485, 306) is a continued-fraction convergent of log(3)/log(2) and (317, 200) and (336, 212) are generalized continued-fraction convergents of log(3)/log(2)).  (For 202 cycles having equal numbers of even and odd elements and K values not equal to the denominator of a generalized continued-fraction convergent of log(3)/log(2) where c is less than or equal to 7999, 3∙log(c) is less than a.)

A histogram of 3∙log(c)-a for the 4896 cycles for c less than or equal to 2999 is;

This distribution has a mean of 11.81311 and a standard deviation of 10.09552.  An upper bound of the minimum |u| or |t| value in a cycle with a (K+L, K) value equal to a generalized continued fraction convergent of log(3)/log(2) can be computed by substituting 3·log(c)+1 for a in the inequality |Λ|<3cac_a2^{-((δ-1)/ζ)K} (Proposition 33) where b=1+cX₀^-1.  Although Proposition (33) is not generally valid for 1-cycles (at least when X₀ is set to the actual minimum |u| or |t| value in the cycle), this gives an X₀ upper bound of 33.0472 for a (K+L, K) value of (11, 7) when c=1 (which is consistent with the data).  For smaller generalized continued-fraction convergents of log(3)/log(2), there are no restrictions on the upper bound when c=1 since according to the inequality, 1/X₀ would be larger than a negative number.  For larger generalized continued-fraction convergents of log(3)/log(2), X₀ would have to be less than 1.  A plot of |Λ|=3cac_a2^{-((δ-1)/ζ)K} solved for X₀ for successive generalized continued-fraction convergents of log(3)/log(2) is;

This gives an X₀ upper bound of 2.1795 for a (K+L, K) value of (970, 612) and an X₀ upper bound of 3.8502 for a (K+L, K) value of (1054, 665) when c=5.  For smaller generalized continued-fraction convergents of log(3)/log(2), there are no restrictions on the upper bound and for larger generalized continued-fraction convergents of log(3)/log(2), X₀ would have to be less than 1.  A plot of the above equation solved for X₀ for successive generalized continued-fraction convergents of log(3)/log(2) is;
 
This gives an X₀ upper bound of 4.7772 for a (K+L, K) value of (1455, 918), an X₀ upper bound of 5.0336 for a (K+L, K) value of (1539, 971), an X₀ upper bound of 2.2587 for a (K+L, K) value of (2108, 1330), and an X₀ upper bound of 1.0939 for a (K+L, K) value of (2593, 1636) when c=7.  For smaller generalized continued-fraction convergents of log(3)/log(2), there are no restrictions on the upper bound and for larger generalized continued-fraction convergents of log(3)/log(2), X₀ would have to be less than 1.  A plot of the above equation solved for X₀ for successive generalized continued-fraction convergents of log(3)/log(2) is;
 
A plot of the above equation solved for X₀ for successive generalized continued-fraction convergents of log(3)/log(2) when c=23 is;

Belaga¹³ calls numbers of the form B_k,l = 2^l - 3^k > 0 "Collatz numbers" and gives a rephrasing of the Diophantine interpretation of the Collatz problem:  no "narrow" Collatz number (i.e., such a (k, l)–Collatz number that 2^l-1 - 3^k <0) could be a divisor of numbers from a certain finite set of natural numbers called the "Collatz (k, l)–corona".  When 2^K+L-3^K is positive and (K+L, K) is a generalized continued-fraction convergent of log(3)/log(2), the Collatz number is narrow.  Since the upper bound of the X₀ values appears to decrease exponentially, evaluating Λ using transcendence theory may be avoidable ((K+L)/K-log(3)/log(2) decreases every other continued-fraction convergent of log(3)/log(2)).  The largest upper bound of the X₀ values occurs when ((δ-1)/(δ^a-1))K (where a=3·log(c)+1) is equal to -s·log(|Λ|) where s is dependent on c and is between 2.56 and 4.27 for c less than or equal to 101 (s gradually increases as c increases [but not monotonically]).  An empirical result is;

(38)  For a given c value, if there are two 3n+c cycles having (L, K) values of (L₁, K₁) and (L₂, K₂) where K₁=K₂ and L₁≠L₂, then either L₁≤K₁ and L₂≥K₂ or L₁≥K₁ and L₂≤K₂.  If there are several such pairs of cycles, say K₁=K₂, L₁≠L₂, K₃=K₄ (K₃≠K₁), L₃≠L₄, and K₅=K₆ (K₅≠K₁, K₅≠K₃), L₅≠L₆, then one of L₁-L₂, L₃-L₄, L₅-L₆ divides the other differences in L values.

Note that for a given c value and number of odd elements, there can be at most three cycles (not counting interrelated cycles) having different L values.  Of the 22982 3n+c cycles for c less than or equal to 9997, there are 24 such pairs of cycles (not counting interrelated cycles) and in every case, neither of the corresponding Collatz numbers is narrow.  An empirical result is;

(39)  For a given c value, if there are two 3n+c cycles have (L, K) values of (L₁, K₁) and (L₂, K₂) where L₁=L₂ and K₁≠K₂, then either L₁≤K₁ and L₂≥K₂ or L₁≥K₁ and L₂≤K₂ (usually).  If there are several such pairs of cycles, then one of the differences in K values divides the other differences in K values (apparently always).

For the 3n+c cycles for c less than or equal to 9997, there are 107 such pairs of cycles (not counting interrelated cycles).  An exception occurs for a pair of cycles for c=145 having (L, K) values of (16, 18) and (16, 32) (16 is approximately equal to 18).  An exception occurs for a pair of cycles for c=1009 having (L, K) values of (56, 37) and (56, 46) (these are the only cycles for this c value, so conditions are less stringent).  A similar exception occurs for a pair of cycles for c=7063 having (L, K) values of (168, 192) and (168, 174).  A third type of exception occurs when there are many such pairs of cycles for a given c value.  For example, when c=6305, the (L, K) values that can be paired are [(24, 12), (24, 20), (24, 28), (24, 36)], [(48, 40), (48, 48), (48, 56)], and [(72, 60), (72, 68), (72, 84)].  Another empirical result is;

(40)  For a given c value, if there are two 3n+c cycles having (K+L, K) values of (K₁+L₁, K₁) and (K₂+L₂, K₂) where K₁+L₁=K₂+L₂, K₁≠K₂, then either L₁≤K₁ and L₂≥K₂ or L₁≥K₁ and L₂≤K₂.  If there are several such pairs of cycles, then one of the differences in K values divides the other differences in K values and one of the differences in L values divides the other differences in L values.

Note that for a given c value and length, there can be at most three cycles (not counting interrelated cycles) having different numbers of odd elements.  Of the 22982 3n+c cycles for c less than or equal to 9997, there are 11 such pairs (not counting interrelated cycles) and in every case, neither of the corresponding Collatz numbers is narrow.  This indicates that if there is a 3n+c cycle having a (K+L, K) value equal to a generalized continued-fraction convergent of log(3)/log(2), then there can't be any other cycles for this c value (other than interrelated cycles) having this K value.  Apparently, Proposition (40) is usually valid (with the same kinds of exceptions as for Proposition (39)) for pairs of cycles where fK₁+gL₁=fK₂+gL₂ (this would account for there being few cycles that aren't interrelated).
 
Another empirical result is;

(41)  The last attachment point in the group of primary, secondary, tertiary, etc., attachment points preceding a primary multiple-jump attachment point is a no-jump or jumped-over attachment point.

A Diophantine equation (similar in form to those derived for primary, secondary, tertiary, etc., attachment points) involving the t value of a multiple-jump attachment point (or at least the odd element at the beginning of the last jump of the multiple-jump) and the t value of the preceding no-jump or jumped-over attachment point can be derived.  

The order of a cycle is defined to be the smallest natural number of the form 3∙2^k greater than or equal to the absolute value of every element of the cycle.  In the following, if all the absolute values of the elements of an extended sequence of a cycle are less than the cycle order, then the extended sequence is further extended backwards until the cycle order has been exceeded (the elements of this portion of the extended sequence will be integers of the form 2ⁱt where t is the odd integer divisible by 3).  The extension order is then taken to be the cycle order.  If the cycle order has already been exceeded between the odd integer divisible by 3 and the attachment point, the order of the extension is defined to be the smallest natural number of the form 3∙2^k greater than or equal to the absolute value of every element of the extended sequence.  The extended sequence is then further extended backwards until this order has been exceeded.  (For the remainder of this article, this will be referred to as being the extended sequence of an attachment point.)  For example, the extended sequence for the c=1 cycle is {-336, -168, -84, -42, -21, -62, -31, -92, -46, -23, -68, -34, -17, -50, -25, -74, -37, -110, -55, -164, -82, -41, -122, -61, -182, -91, -272, -136, -68} and the cycle order and extension order are 384.  An empirical result is;

(42)  For a one-jump attachment point, the cycle order equals the extension order.

Let P denote the primary attachment point having the largest absolute value.  In most 3n+c cycles, 4 times the absolute value of P determines the orders (all the same) of all the extended sequences of the cycle.  Many exceptions are associated with multiple-jump attachments points.  When there is a multiple-jump to P and there is a secondary attachment point, the order of the extended sequence of the primary attachment point may be different from the order of the extended sequence of the secondary attachment point.

Let m be the number of odd elements in an extended sequence up to the attachment point (where the element following an odd element i is temporarily defined to be (3i+c)/2) and e the number of even elements.  For a no-jump attachment point, e is greater than or equal to m.  For a one-jump attachment point where c≤151, e+1 is greater than or equal to m.  For a multiple-jump or jumped-over attachment point where c≤151 and the absolute value of the odd element immediately before the attachment point is greater than order/6, e+2 is greater than or equal to m.  Of the 649 attachment points of the c values less than or equal to 151, e is less than m for only 32 attachment points.  The only c values less than or equal to 151 where e is less than m for a one-jump attachment point are 19, 31, 97, 115, and 139.  In every case, the attachment point equals P and 4|P| determines the order of the extended sequence.  A histogram of the e-m values for the 1807 one-jump attachment points for c values less than or equal to 499 is;

As c increases, the upper bound of m-e appears to gradually increase.  For example, for c=259 and 271, m-e=2 for a one-jump attachment point.  In both cases, the attachment point equals P and 4|P| determines the order of the extended sequence.  For c=187, m-e=3 for a multiple-jump attachment point where the absolute value of the odd element immediately before the attachment point is greater than order/6.  In general, m-e is relatively large for multiple-jump or jumped-over attachment points when there are associated primary, secondary, tertiary, etc., attachment points.

Let max denote the odd integer in the cycle having the largest absolute value and let i denote the odd integer immediately before the primary attachment point following max ((3i+c)/4 equals the primary attachment point).  For example, for a cycle for c=5, max=3397 and the sequence between max and the primary attachment point is {3397, 10196, 5098, 2549, 7652, 3826, 1913, 5744, 2872, 1436} (i=1913 and 3397→2549→1913).  Empirical results are;

(43)  If i≠max and max jumps to i (either by one or multiple jumps), then the primary attachment point is a no-jump attachment point.

(44)  If i≠max and max jumps over i, then the primary attachment point is a one-jump attachment point.

If the primary attachment point is P (that is, i=max) and 4|P| determines the extension order, then the primary attachment point is usually a one-jump attachment point (but may be a multiple-jump attachment point or even a no-jump attachment point). 
 
In this paragraph, cycles containing only one attachment point are discussed.  Also, the order of the extended sequence is required to have been determined by 4|P|.  The sequence vector of such an extended sequence mostly has bilateral symmetry when the number of 2's in the sequence vector (not counting the last element of the sequence vector) is even (usually, it is necessary to disregard the first and last elements of the sequence vector when considering the symmetry, but sometimes it is also necessary to disregard the second or next-to-last elements of the sequence vector).  Since there is only one attachment point, the sequence vector in the interior of the cycle is a 1-2 sequence vector.  Then the sequence vector of the extended sequence is a 1-2 sequence vector except for the first and last elements of the sequence vector.  (The first element of the sequence vector does not indicate that there is an odd sequence element immediately before the extended sequence; it just counts the number of even sequence elements before the first odd sequence element.  A similar situation applies for the last element of the sequence vector.)  The sequence vectors of the extended sequences of the interrelated cycles for c=13 are (4, 1, 1, 1, 1, 2, 2), (3, 2, 1, 1, 1, 2, 1, 2), (4, 1, 2, 1, 1, 2, 1, 1, 2), (4, 1, 1, 2, 1, 2, 1, 1, 1, 2), and (5, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2).  The sequence vectors of the extended sequences of the interrelated cycles for c=37 and a cycle length of 9 are (6, 2, 1, 2, 1, 2) and (5, 1, 2, 2, 1, 1, 2).  The sequence vectors of the extended sequences of the interrelated cycles for c=47 and a cycle length of 11 are (4, 1, 1, 1, 2, 2, ), (4, 2, 1, 1, 2, 1, 2), (6, 1, 2, 1, 2, 1, 1, 2), and (7, 1, 1, 2, 2, 1, 1, 1, 2).  The sequence vectors of the extended sequences of the interrelated cycles for c=101 and a cycle length of 10 are (6, 2, 1, 2, 2) and (5, 1, 2, 2, 2).  In all of these cases, it is possible to list the interrelated cycles so that the number of 1's between the pair of 2's (if there is a pair) decreases by one and the extension order remains the same or increases for each successive interrelated cycle. 

Generalized Dead Limbs

Consider a 3n+1 sequence where the fourth-to-last element of the sequence (denoted by i) is odd, the third-to-last element of the sequence is divisible by 8, and the sequence has been extended backwards (with any restrictions on the path taken), until an odd integer divisible by 3 has been encountered.  The order of this sequence is defined to be the smallest natural number of the form 3∙2^k greater than or equal to the element in the sequence with the largest absolute value.  The sequence is then further extended backwards until the order is exceeded.  Note that if the absolute value of i is greater than the order divided by 6, then the absolute value of the third-to-last element of the sequence (3i+1) is greater than the order divided by 2, that is, the third-to-last element of the sequence (along with other elements of the sequence) determines the order.  When the absolute value of i is greater than the order divided by 6, these sequences will be referred to as "generalized dead limbs" (ordinary dead limbs will be defined later).  In the following, the element after an odd element j is temporarily defined to be (3j+1)/2 (for the purpose of determining lengths).  Let e denote the number of even elements in a generalized dead limb, m the number of odd elements, and l the number of elements.  Also, generalized dead limbs containing duplicated cycles are excluded.  An empirical result is;

(45)  For a given number of odd elements in a generalized dead limb, the number of even elements is mostly fixed; the number of even elements can deviate by at most 1.

For example, for m=1, e can equal 4 or 3.  (The (e, m) value will be denoted by ((4, 3), 1).)  (e, m) values for generalized dead limbs are ((4, 3), 1), ((5, 4), 2), ((5, 4), 3), ((6, 5), 4), ((6, 5), 5), ((7, 6), 6), ((8, 7), 7), ((8, 7), 8), ((9, 8), 9), ((9, 8), 10), ((10, 9), 11), ((11, 10), 12), ((11, 10), 13), ((12, 11), 14), ((12, 11), 15), ((13, 12), 16), ((13, 12), 17), ((14, 13), 18), ((15, 14), 19), ((15, 14), 20), ((16, 15), 21), (16, 15), 22), ((17, 16), 23), ((18, 17), 24), ....  An empirical result is;

(46)  The adjusted (l, m) values (where l is decremented by 4 or 3) of generalized dead limbs include the generalized continued-fraction convergents of log(3)/log(2).

The (l, m) value of a 3n+1 cycle is a generalized continued-fraction convergent of log(3)/log(2) (based on empirical evidence).  (Assuming the factor of 1.000001 in Simons and de Weger's inequality can be reduced to 1.0, the (l, m) value of a 3n+1 cycle [n>0] is a continued-fraction convergent of log(3)/log(2).)  For example, the (l, m) value of the 3n+1 cycle {-34, -17, -25, ..., -68} is (11, 7) and the (l, m) value of the extended sequence {-336, -168, -84, ..., -68} (a generalized dead limb containing the cycle) is (19, 10).  An empirical result is;

(47)  The (l, m) value of a generalized dead limb minus the (l, m) value of a generalized continued-fraction convergent of log(3)/log(2) is the (l, m) value of a shorter generalized dead limb.

Let f denote the mapping of the number of odd elements in a generalized dead limb to the smaller of the two possible numbers of even elements in the limb.  Attachment points were previously defined for cycles.  The definition of attachment points for generalized dead limbs is the same; the attachment points are just not necessarily in cycles.  An empirical result is;

(48)  The number of even elements in a generalized dead limb containing x primary attachment points is optimal in that the number of even elements up to the yth primary attachment point, y<x, is greater than or equal to f(m) where m is the number of odd elements up to the yth primary attachment point.

Weibull probability distributions are used to model birth and decay processes and have a scaling and shaping parameter.  For the generalized dead limbs where the odd elements divisible by 3 are less than a fixed amount and the orders are less than a fixed amount, the number of even elements (for a given number of odd elements) up to the yth primary attachment point has a Weibull probability distribution.  For example, for generalized dead limbs (in the 3n-1 sequence, n>0) where the odd elements divisible by 3 are less than or equal to 99999999, the orders are less than or equal to 0x30000000, x=5, and y=3, the numbers of generalized dead limbs where m=10 and e-8 equals 0, 1, 2, ..., 11 are 62, 1246, 2939, 3593, 3070, 1850, 1014, 415, 168, 67, 31, and 2 respectively (f(10)=8).  The parameters of the Weibull probability distribution for this data are 889.0686 (with a 95% confidence interval of (338.0, 2338.6) and 0.6140 (with a 95% confidence interval of (0.4, 1.0).  A plot of the data is;


When c>1, the behavior of such sequences approaches that of c=1 in a limiting sense, that is, for a fixed c value and a sufficiently large order, the sequences have the same (l, m) values as for c=1 (an empirical result).

For the 3n+1 cycle and the extended sequence of P (-68), (8, 3) ((19, 10) [the (l, m) value of the extended sequence] minus (11, 7) [the (l, m) value of the cycle and a generalized continued-fraction convergent of log(3)/log(2)]) is the (l, m) value of a shorter generalized dead limb.  Similar cycles occur in general 3n+c sequences.  For the 3n+5 cycle and the extended sequence of P where P=1562, (5, 1) ((32, 18) [the (l, m) value of the extended sequence] minus (27, 17) [the (l, m) value of the cycle and a generalized continued-fraction convergent of log(3)/log(2)]) is the (l, m) value of a shorter generalized dead limb.  For another 3n+5 cycle and extended sequence having (l, m) values of (27, 17) and (32, 18) respectively, the difference is (5, 1).  For the 3n+13 cycle and the extended sequence of P where P=454, (5, 1) ((13, 6) [the (l, m) value of the extended sequence] minus (8, 5) [the (l, m) value of the cycle and a continued-fraction convergent of log(3)/log(2)]) is the (l, m) value of a shorter generalized dead limb.  For four other 3n+13 cycles having (l, m) values of (8, 5), the differences are (6, 2), (8, 3), (9, 4), and (11, 5) (all (l, m) values of shorter generalized dead limbs).  For the 3n+13 cycle and the extended sequence of P where P=1048, (5, 1) ((29, 16) [the (l, m) value of the extended sequence] minus (24, 15) [the (l, m) value of the cycle and a generalized continued-fraction convergent of log(3)/log(2)]) is the (l, m) value of a shorter generalized dead limb.  For the 3n+23 cycle and the extended sequence of P where P=-23704, (12, 6) ((31, 18) [the (l, m) value of the extended sequence] minus (19, 12) [the (l, m) value of the cycle and a generalized continued-fraction convergent of log(3)/log(2)]) is the (l, m) value of a shorter generalized dead limb.  For six other 3n+23 cycles having (l, m) values of (19, 12), the differences are (12, 6), (8, 3), (9, 4), (5, 1), (7, 2), and (5, 1) (all (l, m) values of shorter generalized dead limbs).  For two of these cycles, 4|P| does not determine the order of the extended sequence (this frequently occurs for interrelated cycles).  For the 3n+29 cycle containing the primary attachment points 788552, 194366, 79082, 73934, 41606, and 18794, (7,2) ((72, 43) minus (65, 41) [a continued-fraction convergent of log(3)/log(2)]) is the (l, m) value of a shorter generalized dead limb.  For the 3n+29 cycle containing the primary attachment points 1707830, 1441016, 360254, 577232, 36902, 20306, and 7622, (10, 4) ((75, 45) minus (65, 41)) is the (l, m) value of a shorter generalized dead limb.  For the 3n+71 cycle containing the primary attachment points 41360 and 17042, (6, 2) ((33, 19) minus (27, 17)) is the (l, m) value of a shorter generalized dead limb.  For four other 3n+71 cycles having (l, m) values of (27, 17), the differences are (6, 2), (5, 1), (5, 1), and (5, 1) (all (l, m) values of shorter generalized dead limbs).  For two of these cycles, 4|P| does not determine the order of the extended sequence.  For the 3n+139 cycle containing the primary attachment point 16472, (20, 11) ((31, 18) minus (11, 7)) is the (l, m) value of a shorter generalized dead limb.  For twenty-three other 3n+139 cycles having (l, m) values of (11, 7), the differences are (13, 6), (13, 7), (10, 5), (9, 4), (11, 5), (12, 6), (6, 2), (9, 4), (5, 1), (8, 3), (9, 4), (6, 2), (11, 5), (5, 1), (8, 3), (7, 2), (8, 3), (5, 1), (6, 2), (5, 1), (7, 2), (5, 1), and (5, 1).  Except for (13, 7), these differences are (l, m) values of shorter generalized dead limbs.  (The permissible (l, m) values for a generalized dead limb containing seven odd elements are (15, 7) and (14, 7).)  For five of these cycles, 4|P| does not determine the order of the extended sequence.  Similar 3n+c cycles and extended sequences occur for c=311, 343, 355, 467, ....

Hypothetical 3n+1 Cycles

Properties of 3n+1 cycles can be inferred from those of general 3n+c cycles that are in generalized dead limbs and have (l, m) values of generalized continued-fraction convergents of log(3)/log(2).  That a 3n+1 cycle having an attachment point should be in a generalized dead limb is to be expected since most 3n+c cycles having attachment points are in generalized dead limbs and the 3n+1 sequence is comparatively "well-behaved".  Catalan's conjecture (proved by Mihăilescu¹⁴) states that the only natural number solutions of x^a-y^b=1 are x=3, a=2, y=2, and b=3.  The only solution of 2^l-3^m=-1 is then (3, 2).  Also, |2^l-3^m| increases monotonically (apparently) for (l, m) values that are generalized continued-fraction convergents of log(3)/log(2) (excluding (2, 1), (4, 2), (6, 4), and (9, 6)), so there shouldn't be any more 3n+1 cycles where |2^l-3^m|=1.  Study of 3n+c cycles where c properly divides |2^l-3^m| and the (l, m) values are relatively large should yield expected properties of any other 3n+1 cycles (the number of odd elements in a hypothetical 3n+1 cycle [n>0] has been confirmed to be very large).  A c value of 467 (and several others) fulfills these and the above requirements.  When c=467, there are fifteen primitive cycles having (l, m) values of (84, 53) (a continued-fraction convergent of log(3)/log(2)) and these cycles have 120 attachment points.  Empirical results are;

(49)  When the domain of the absolute values of the u values and the absolute values of their proxies is sufficiently small, the (l, m) value of a 3n+c cycle is a generalized continued-fraction convergent of log(3)/log(2), the cycle is in a generalized dead limb, and the (l, m) value of the extended sequence of P minus the (l, m) value of the cycle is the (l, m) value of a shorter generalized dead limb.

(50)  When the domain of the absolute values of the u values and the absolute values of their proxies is sufficiently small, all the extended sequences of a 3n+c cycle have the same order, the sign does not change in an extended sequence, and there are more even elements in an extended sequence than odd elements (the only possible (e, m) values of the extended sequence of P are then ((4, 3), 1), ((5, 4), 2), ((5, 4), 3), ((6, 5), 4), ((6, 5), 5), ((7, 6), 6), ((8, 7), 7), ((8, 7), 8), and ((9, 8), 9)).

(51)  When the domain of the absolute values of the u values and the absolute values of their proxies is sufficiently small, there are no jumped-over attachment points (so that the attachment points are either primary or secondary attachment points), P is a one-jump attachment point, one-jump or multiple-jump attachment points are primary attachment points, secondary attachment points are no-jump attachment points, multiple-jump attachment points do not have associated secondary attachment points, and a multiple-jump attachment point is preceded by a no-jump attachment point (possibly a primary attachment point).

Here, "sufficiently small" means being on the verge of becoming too small.  For a c=467 cycle, the plot of the |u| values and the absolute values of the corresponding proxy values versus their domain is;

(The x values have been scaled up by a factor of 10⁸ and all but three data points are shown.  No shape-preserving interpolation of data points is done.)   The domain is (-2.390748e-007, 4.432486e-007). 

As shown previously, the number of even elements in the extended sequence is sometimes smaller than the number of odd elements for a one-jump attachment point (especially when the attachment point is P).  For two of the cycles for c=467, the number of even elements in an extended sequence is one less than the number of odd elements and this occurs for one-jump attachment points where the attachment point is P.  For four of the 71 one-jump attachment points for the c=467 cycles, t≡u(mod 2^j) where j is greater than the number of odd elements in the jump (in every case, locmin is not equal to u).  For three of the 15 cycles, there is not a one-jump attachment point where j equals the number of odd elements in the jump.

When c=311, there are two primitive cycles having an (l, m) value of (57, 36) (a generalized continued-fraction convergent of log(3)/log(2)).  There are 11 attachment points and these attachment points have the same properties as the c=467 cycles.  For one of the cycles, the number of even elements in an extended sequence equals the number of odd elements and this occurs for a one-jump attachment point where the attachment point is P.  For one of these cycles, there is not a one-jump attachment point where j equals the number of odd elements in the jump.  When c=311, there are seventeen primitive cycles having an (l, m) value of (38, 24) (a generalized continued-fraction convergent of log(3)/log(2)).  There are 66 attachment points and these attachment points have the same properties as the c=467 cycles except that there are two jumped-over attachment points (in different cycles).  For two of the cycles the number of even elements in an extended sequence is one less than the number of odd elements and this occurs for one-jump attachment points where the attachment point is P.  For fourteen of the 41 one-jump attachment points, t≡u(mod 2^j) where j is greater than the number of odd elements in the jump (usually locmin is not equal to u).  For seven of the cycles, there is not a one-jump attachment point where j equals the number of odd elements in the jump.  The domains of the absolute values of the u values and the absolute values of their proxies are somewhat larger than the domains of the c=467 cycles having (l, m) values of (84, 53).

When c=343, there is a primitive cycle having an (l, m) value of (149, 94) (a generalized continued-fraction convergent of log(3)/log(2)).  The domain of the absolute values of the u values and the absolute values of their proxies is (-6.216047e-007, 1.015725e-006), somewhat larger than the domains of the c=467 cycles having (l, m) values of (84, 53).  There are 15 attachment points and these attachment points have the same properties as the c=467 cycles having (l, m) values of (84, 53) except that P is a no-jump attachment point.  For all eight one-jump attachment points, t≡u(mod 2^j) where j is less than or equal to the number of odd elements in the jump.  For three of these attachment points, j equals the number of odd elements in the jump.

When c=1091, there is a primitive cycle having an (l, m) value of (130, 82) (a generalized continued-fraction convergent of log(3)/log(2)).  The domain of the absolute values of the u values and the absolute values of their proxies is (-3.446479e-007, 8.889605e-007), somewhat larger than the domains of the c=467 cycles having (l, m) values of (84, 53).  There are 12 attachment points and these attachment points have essentially the same properties as the c=467 cycles having (l, m) values of (84, 53) except that there is a jumped-over attachment point and the number of even elements in an extended sequence for a multiple-jump attachment point is equal to the number of odd elements.  For one of the 7 one-jump attachment points, t≡u(mod 2^j) where j is greater than the number of odd elements in the jump (locmin is not equal to u).  For  two of these attachment points, j equals the number of odd elements in the jump.

When c=2507, there are three primitive cycles having an (l, m) value of (76, 48) (a generalized continued-fraction convergent of log(3)/log(2)) and the domains of the absolute values of the u values and the absolute values of their proxies are (-1.018299e-006, 2.239814e-006), (-2.209624e-007, 7.065484e-007), and (-1.465981e-006, 2.967221e-006).  In one of the cycles, P is a no-jump attachment point.  In another cycle, there is a jumped-over attachment point and the number of even elements in an extended sequence for a one-jump attachment point (that of P) is equal to the number of odd elements.  In the remaining cycle, there is a jumped-over attachment point preceding a multiple-jump attachment point.  For one of the 11 one-jump attachment points for these cycles, t≡u(mod 2^j) where j is greater than the number of odd elements in the jump (locmin is not equal to u).  For one of these cycles, j equals the number of odd elements in the jump for three of the 5 one-jump attachment points.  Although the domains appear to be small, they are not small enough for this large of a c value to exhibit all of the above properties; many anomalies occur.  For comparison, when c=29, there are two primitive cycles having an (l, m) value of (65, 41) (a continued-fraction convergent of log(3)/log(2)) and the domains of the absolute values of the u values and the absolute values of their proxies are (-1.136015e-005, 2.425506e-005) and (-2.555711e-005, 3.583320e-005).  There are 15 attachment points and these attachment points have the same properties as the c=467 cycles having (l, m) values of (84, 53) (except there are no anomalies).

When c=5, 13, 23, 71, 139, and 355, there are cycles having (l, m) values of generalized continued-fraction convergents of log(3)/log(2) that also have some properties of the c=467 cycles having (l, m) values of (84, 53) .  Also, when c=407, cycles having an (l, m) value of (44, 28) (almost equal to (46, 29) [a generalized continued-fraction convergent of log(3)/log(2)]) have some properties of the c=467 cycles.  

This is currently the extent to which 3n+c cycles with attachment points have been investigated.  In the remainder of the article, 3n+c cycles not having attachment points are discussed.  

The remaining step is to find the smallest order for which a given e value occurs.  One-half of the limbs in S consist of 1 element, about 1/3 of the limbs in S consist of two elements, about 1/12 of the limbs in S consist of four elements, about 1/72 of the limbs in S consist of five elements, about 1/36 of the limbs in S consist of seven elements, about 5/576 of the limbs in S consist of nine elements, about 7/864 of the limbs in S consist of ten elements, about 77/7776 of the limbs in S consist of twelve elements, about 13/15552 of the limbs in S consist of fourteen elements, and about 815/186624 of the limbs in S consist of fifteen elements.  The denominators of these fractions are a power of two times a power of three.  Denote a fraction by f.  The number of limbs in S (for an order of 3∙2^k, k>2) is 2^k-1.  Denote the number of limbs in S of a given length by m.  For limb lengths of 2, 4, or 5, 0≤[2^k-1f]+1-m≤1, for limb lengths of  7, 9, or 10, -1≤[2^k-1f]+1-m≤2, and for limb lengths of 12, 14, or 15, -2≤[2^k-1f]+1-m≤3, where the brackets denote the greatest integer function.  About 2/3 of  the limbs in S with a length of 1 are alive, about 1/2 of the limbs in S with a length of 2 are alive, about 1/2 of the limbs in S with a length of 4 are alive, about 1/6 of the limbs in S with a length of 7 are alive, about 1/3 of the limbs in S with a length of 9 are alive, about 25/154 of the limbs in S with a length of 12 are alive, and about 1/4 of the limbs in S with a length of 14 are alive.  Denote one of these fractions by f' and denote the number of live limbs in S of a given length by m'.  For limb lengths of 1, 0=[2^k-1ff']-m', for limb lengths of 2, 4, 7, or 9, 0≤[2^k-1ff']-m'≤1, and for limb lengths of 12 or 14, -1≤[2^k-1ff']-m'≤2.  For longer limb lengths, the portion of the limbs in S is similarly fixed.  The a+b values of the live limbs in S with lengths of 2, 4, 7, 9, 12, 14, ..., are 1, 2, 3, 4, 5, 6, ...., respectively.  For every value of j, j=1, 2, 3, ..., there is a relatively small odd natural number g, a relatively small natural number h, and a natural number i approximately equal to j such that 2^k-1(g/(2^h 3ⁱ)) closely approximates the number of live limbs in S with the corresponding length.  For live limbs in S with lengths of 17, 20, 22, 25, 27, 30, and 33, the g values are 233, 695, 6349, 3791, 2831, 3425, and 7359 respectively, the h values are 11, 11, 10, 12, 13, 8, and 13 respectively, and the i values are 5, 8, 10, 9, 9, 12,  and 13 respectively.  (These values aren't unique; similar g and h values can be found when i is set to j.)  For some limb lengths, live limbs associated with certain sequence vectors occur more frequently than live limbs associated with other sequence vectors.  For example, for a limb length of 12, there is a group of live limbs in S associated with the sequence vector giving the largest e value where the probability of occurrence is 17/(2⁶∙3⁵) and there is a group of live limbs in S associated with the sequence vector giving the smallest e value where the probability of occurrence is 1/(2³∙3⁵) (the sum of these probabilities is 25/(2⁶∙3⁵)).  For a limb length of 17, there is one group of live limbs in S associated with two sequence vectors (one of which is the sequence vector giving the largest e value) where the probability of occurrence is 1/(2¹⁰∙3¹), another group of live limbs in S associated with two other sequence vectors (one of which is the sequence vector giving the smallest e value) where the probability of occurrence is 13/(2¹⁰∙3⁵), and another group of live limbs in S associated with the remaining sequence vector where the probability of occurrence is 5/(2¹¹∙3³) (the sum of these probabilities is 233/(2¹¹∙3⁵)).  For a limb length of 22, the probabilities of occurrence for live limbs in S in the different groups are 1147/(2⁸∙3¹⁰), 5/(2¹³∙3³), and 13/(2¹³∙3⁵).  For a limb length of 25, the probabilities of occurrence for live limbs in S in the different groups are 1237/(2¹¹∙3⁹), 853/(2¹²∙3⁹), and 29/(2⁸∙3⁹).  For a limb length of 27, the probabilities of occurrence for live limbs in S in the different groups are 2567/(2¹³∙3⁹) and 11/(2¹⁰∙3⁸).  For a limb length of 30, the probabilities of occurrence for live limbs in S in the different groups are 131/(2¹²∙3⁷), 107/(2¹²∙3⁸), 865/(2⁹∙3¹²), 215/(2⁹∙3¹¹), and 185/(2¹⁰∙3¹¹).  For limb lengths of 14, 20, or 33, there is only one group of live limbs in S associated with the different sequence vectors.  The differences between the estimated number of live limbs in S using these probabilities (where rounding is done) and the actual number of live limbs in S for different orders and limb lengths (denoted by L) when c=1 are;

        k=12   13   14   15   16   17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32   33
L=2       0     1     0     1     0     1     0     1     0     1     0     1     0     1     0     1     0     1     0     1     0     1
     4       0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     7       0     0     0     0     1   -1     0     0     0     0     0     0     0    -1    1    -1     0     0     0     0     0     0
     9       0     0     1     0     0     1     0     1     0     0     0     0     1     0     0     1     0     0     0     0     1     0
   12       0     0     0     0     0   -1     1   -1     1    -1   -1     0     0     0    -1    -1    0     0      0     0     0     0
   14      -1    0     0   -1     0     0    -1     0     0     0     0     0     0     0     0     0   -1     0     -1     0    -1    0
   17       0     1     1     1     0     1     0     1   -1     1     0     1     0     2     1     1     0     1     0     0     1     0
   20       0   -1     0    -1     1   -1     1     0     1    -1     1   -1     1     0     1    -1     0   -1     1    -1     1   -1
   22       0   -1     1     0    -3   -1    -2    0    -1    -1   -1   -2    -3   -2    -1   -3    -2     0     1     1     1     2
   25       0     0     0     1     2     1   -2     3     0     5     4     4     1     2     0     5     0    -1   -4     1     1     6
   27       0     0     0     0     1   -1     1     2     3    -1    0     4    -1     1     2     1     5     2     1     1     0     0
   30       0     0     0     0     0     1     3   -3     1     0    -2   -2    -1    1    -2    -2   -2     1     0     1     1     4
   33       0     0     0     0     0     0     0    0    -1     1    -1    1     0    -4     1     0     2    -3     2   -5      0   -7     
 
For every order and limb length, the difference between the number of live limbs in S associated with one sequence vector in a group and the number of live limbs in S associated with another sequence vector in the group is at most 1.  (This appears to still be true for live limbs in A, B, C, or D or any other similarly constructed subset of S where the last elements in the limbs are congruent to an odd natural number j modulo 2ⁱ, j<2ⁱ , i<k).

The above property can be used to estimate the smallest order for which a live limb in S with a given length occurs.  For example, assuming that the upper bound of the difference between the estimated and actual number of live limbs with a length of 25 (where the difference is given by the formula [2^k-1(3791/(2¹²∙3⁹))+0.5]-m') is 6 (a somewhat dubious proposition) gives m'≥1 when k=19 (when c=1, the smallest order for which a live limb with a length of 25 occurs is 3∙2¹⁷).  The n value for a limb length of 25 is 6487 and the largest e value is 811694, so n(order/2) is much greater than ce, c>0, n>0, when order=3∙2¹⁹.  The difficulty with this approach is determining the upper bound of the difference between the estimated and actual number of live limbs.

When c>1, there are relatively few limbs in S having lengths other than 1, 2, 4, 5, 7, 9, 10, 12, 14, 15, 17, 18, 20, 22, 23, 25, ....  When k is sufficiently large for a given limb length and c value greater than 1, the limbs in S have many of the above properties.

Another Kind of Least-Residue Tree

In this section, c is restricted to being 1 or -1.  Consider a tree where the order is of the form 3∙2^k and no element of a limb is greater than or equal to the order or divisible by 3.  The first element of a limb must be even and greater than the order divided by 2, the last element of a limb must be odd and greater than the order divided by 3, and every limb must contain at least one element that is divisible by 8.  (As for a limb in S of a least-residue tree, a limb cannot attach to its interior or the interior of another limb.)  The possible lengths of these limbs are 12, 14, 15, 17, 18, 20, 22, 23, ....  The lengths of the limbs are the same as those for a limb in S of a least-residue tree (except there are no lengths of 1, 2, 4, 5, 7, or 9) and the X and Y values are the same.  Denote the first element of a limb by y, the last element of a limb by i, (3i+c)/2 by z, and y-z by x.  As before, x and y satisfy the Diophantine equation ny-dx=ce where n=X-Y, d=X, and the e values are fixed for a given limb length.  Some e values are;

length=12, e=1202
length=14, e=3862
length=15, e=7724, 5780
length=17, e=24196, 22738, 18850, 18364, 16906, 16258, 14530, 14476, 13378, 13018
length=18, e=48392, 45476, 37700, 36728, 33812, 32516, 29060, 28952, 26756, 26036

For a given limb length, the largest e value is larger than the largest e value for a live limb in S of a least-residue tree (when not all of the limbs are dead for that length) and the smallest e value (when there is more than one e value) is smaller than the largest e value for a live limb in S of a least-residue tree (when not all the limbs are dead for that length).  The lengths of the sequence vectors for limb lengths of 12, 14, 15, 17, 18, 20, 22, and 23 are 5, 6, 6, 7, 7, 8, 9, and 9 respectively.  The largest elements in the sequence vectors for limb lengths of 12, 14, 15, 17, 18, 20, 22, and 23 are 3, 3, 3, 4, 4, 5, 5, and 5 respectively.  For some limb lengths, limbs associated with certain sequence vectors occur more frequently than limbs associated with other sequence vectors.  For example, for a limb length of 17, the number of  limbs associated with two sequence vectors is about [2^k-1(295/(2⁹∙3⁶))+0.5], the number of limbs associated with five other sequence vectors is about [2^k-1(5/(2¹⁰∙3¹))+0.5], the number of limbs associated with two other sequence vectors is about [2^k-1(5/(2⁹∙3³))+0.5], and the number of limbs associated with the remaining sequence vector is about [2^k-1(13/(2¹⁰∙3⁵))+0.5].  For every order and limb length, the difference between the number of limbs associated with one sequence vector in a group and the number of limbs associated with another sequence vector in the group is at most 2.  For some limb lengths, the upper bound of the maximum y value to order ratio associated with a group of sequence vectors is different from the upper bounds associated with other groups of sequence vectors.  (These groups of sequence vectors are the same as those giving different frequencies of occurrence of limbs.)  For example, for a limb length of 17, the upper bound of the maximum y value to order ratio associated with two sequence vectors is 512/729, the upper bound of the maximum y value to order ratio associated with five other sequence vectors is 2/3, the upper bound of the maximum y value to order ratio associated with two other sequence vectors is 16/27, and the upper bound of the maximum y value to order ratio associated with the remaining sequence vector is 128/243.  For limb lengths of 12, 14, 15, 17, 18, 20, 22, and 23, the number of limbs can be approximated by [2^k-1f+0.5] where f equals 17/(2⁵∙3⁵), 13/(2⁸∙3⁵), 353/(2⁸∙3⁶), 1057/(2⁹∙3⁶), 695/(2⁹∙3⁷), 6733/(2¹⁰∙3⁷), 9817/(2¹³∙3⁶), and 56915/(2¹⁴∙3⁷) respectively.  The number of limbs in one of these trees is approximately equal to the order divided by 144.

M-Cycles 

In their 2005 article, Simons and de Weger gave bounds for Λ=(K+L) log 2 - K log 3 (where K is the total number of odd elements and L is the total number of even elements in the m-cycle).  A corollary proved is 0<Λ<m/x_min.  For a limb in S of a least-residue tree (with a length of 3a+2b) to attach to itself at its beginning, this gives x_min<(a+1)/((2a+b+1) log 2 - (a+b) log 3).  This upper bound for the minimum is about twice as large as the minimum given by Halbeisen and Hungerbühler's formula.  In the next two sections, the upper bound of the minimum will be computed when it's not computationally feasible to compute the minimum.

The Minimum in a Collatz Cycle

In this section, c is restricted to being 1 or -1.  The results in this section are mostly empirically derived.  The 1-2 sequence vector (of a limb in S having a length of l and containing m odd elements) corresponding to a small e value (possibly the smallest) can be constructed from ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l, by rotating (if necessary) the vector so that a (1, 1) is at the beginning of the vector and converting (1, 0)'s to 2's.  The last element of the sequence vector will be a 2 and must be changed to a 1 to match the sequence vector corresponding to the smallest e value.  (Note that this rotation of the parity vector does not correspond to the parity vector of a limb in S [since the first element must be even] and that the first element of a sequence vector just indicates that the first element of the limb is even [and not that there is an odd natural number in the 3n+c sequence immediately preceding it].  If a cycle were formed by a limb attaching to itself at its beginning, the first element of the sequence vector would be changed to a 2 [and the 2 would indicate that there were 2 even elements between odd elements in the sequence].)  For example, for a limb length of 20, the parity vector must be rotated right by 5 positions for the resulting sequence vector to match the sequence vector corresponding to the smallest e value.  (The possible right-rotations for a limb length of 20 are 0, 5, and 10.  In general, the difference between successive right-rotations is 3 or 5.)  The required right-rotations of the parity vector for limb lengths of 2, 4, 7, 9, 12, 14, 17, 20, 22, 25, 27, 30, 33, 38, and 43 are 0, 0, 0, 0, 0, 0, 0, 5, 11, 0, 11, 0, 5, 0, and 0 respectively.  If the un-rotated parity vector gives an e value for a limb in S, then the e value is the smallest for a limb in S and all the possible right-rotations give e values for limbs in S.  For limb lengths of 20, 22, 27, 33, 35, 40, 45, 48, 51, 53, and 56, the un-rotated parity vector gives e values too small to be a limb in S.  In these cases, a right-rotation of 5 gives an e value (usually not the smallest) for a limb in S.

A rotation of the parity vector ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l, so that a (1, 0) is at the beginning of the vector does not correspond to a live limb in S since the second element of a live limb in S is odd.  A rotation of the parity vector so that a (0, 1) is at the beginning of the vector corresponds to a live limb in S, but except for limb lengths of 7, 12, 17, 30, 43, ..., the e values given are too small.  For a given limb length, live limbs in S occur for only one right-rotation of the parity vector ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l.  (The resulting sequence vectors for limb lengths of 7, 12, 17, 30, and 43 are (1, 2, 1), (1, 2, 1, 2, 1), (1, 2, 1, 2, 1, 2, 1), (1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1), and (1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1) respectively.  [For example, for a limb length of 43, the parity vector is 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, and 1.  Changing the last 1 and the first 0 (a (1, 0)) to a 1 and the other (1, 0)'s to 2's gives the sequence vector.]  These sequence vectors have bilateral symmetry and are the sequence vectors corresponding to the smallest e values.)
 
For limb lengths less than or equal to 51, no limb in S can have the parity vector ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l, if l and m are not relatively prime (51 is the largest limb length where it is computationally feasible to determine this [at least by an exhaustive search up to relatively large orders of least-residue trees]).  For limb lengths of 7, 12, 17, 30, and 43, l and m are relatively prime and there are limbs in S having the parity vector ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l, (the (l, m) values for limb lengths of 12 and 30 [(8, 5) and (19, 12) respectively] are continued-fraction convergents of log(3)/log(2)).  For limb lengths of 20, 22, 27, 33, 40, and 48, l and m are relatively prime, but there are no limbs in S having the parity vector ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l.

For a live limb in S, the second sequence vector value is a 2 if the limb length is greater than 4 (so that the parity vector must begin with a (0, 1, 0)).  Let v₁, v₂, v₃, ..., v_l be a right-rotation of the parity vector ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l, such that v₁=0, v₂=1, and v₃=0.  Let s₁=1.0 and s_i+1=(3/2)s_i if v_i=1 or (1/2)s_i otherwise, i=1, 2, 3, ..., l-1.  Let r denote gcd(l, m).  Requirements for a live limb in S (with a sufficiently large beginning element y) to have this right-rotated parity vector are that (1) twice the minimum s value be greater than the maximum "odd" s value for all possible right-rotations of the parity vector where v₁=0, v₂=1, and v₃=0, (2) the difference in successive right-rotations of the parity vector and the difference in successive indices (modulo l/r) of the minimum s value must be the same for all possible right-rotations of the parity vector where v₁=0, v₂=1, and v₃=0, and (3) the difference in the indices (modulo l/r) of the minimum and maximum "odd" s values must be the same for all possible right-rotations of the parity vector where v₁=0, v₂=1, and v₃=0.  Limb lengths satisfying the first condition for m values less than or equal to 100 are 7, 12, 17, 20, 25, 30, 38, 43, 51, 56, 61, 74, 87, 92, 105, 123, 136, 211, and 242.  Of the first 8192 limb lengths, 20 is the only limb length that satisfies the first and third conditions, but not the second condition.  Of the first 8192 limb lengths, 56 is the only limb length that satisfies the first condition, but not the second and third conditions.  (l, m) values satisfying the first condition are (5, 3), (8, 5), (11, 7), (13, 8), (16, 10), (19, 12), (24, 15), (27, 17), (32, 20), (35, 22), (38, 24), (46, 29), (54, 34), (57, 36), (65, 41), (76, 48), (84, 53), (130, 82), (149, 94), (168, 106), (233, 147), (252, 159), (317, 200), (336, 212), (401, 253), (420, 265), (485, 306), (504, 318), (569, 359), (970, 612), (1054, 665), (1455, 918), (1539, 971), (2108, 1330), (2593, 1636), (3162, 1995), (3647, 2301), (4216, 2660), (4701, 2966), (5270, 3325), (5755, 3631), (6324, 3990), (6809, 4296), 7378, 4655), (7863, 4961), (8432, 5320), (8917, 5626), (9486, 5985), (9971, 6291), (10540, 6650), (11025, 6956), (11594, 7315), (12079, 7621), (12648, 7980), ....  Except for (13, 8), (32, 20), (35, 22), and (54, 34), these are generalized continued-fraction convergents of log(3)/log(2) (all the convergents except (1, 1), (2, 1), (3, 2), (4, 2), (6, 4), and (9, 6) are included). 
   
The Minima in Collatz Cycles and Prime Gaps

In this section, c is restricted to being 1 or -1 and Halbeisen and Hungerbühler's formula is used to compute the minimum in a cycle formed by a limb in S of a least-residue tree attaching to itself at its beginning (this assumes the cycle has the above parity vector).  l denotes the number of elements in the cycle and m denotes the number of odd elements.  The absolute value of 2^l-3^m is not used when computing the "minimum".  The minima where m is a multiple of a natural number d increase and decrease in a regular fashion.  For example, the truncated minima for m=5, 10, 15, ..., 65 are 24, 24, 24, 24, 24, -19, -27, -35, -48, -70 -111, -219, and -1004 respectively, the truncated minima for m=70, 75, 80, ..., 130 are 463, 204, 137, 108, 88, -58, -70, -86, -111, -147, -219, -361, and -1004 respectively, and the truncated minima for m=135, 140, 145, ..., 195 are 1561, 463, 281, 204, 162, 137, -111, -133, -165, -219, -295, -462, and -1004 respectively.  In general, the minima for d =5 have a roughly saw-tooth shaped curve with a period of 13 or 14.  Some long periods (and large minima) occur for d=7, 12, 17, 19, 22, and 29; the periods are 10 or 11, 51 or 52, 17 or 18, 8 or 9, 7 or 8, and 27 or 28 respectively.  The "superposition" of these quasi-periodic "waves" is of importance in determining the largest minima.  For example, a large truncated minimum (for c=-1) of -3664765 occurs when m=665=5∙7∙19.

The periods for prime m values less than 41346 and of the form 12k-1, k=1, 2, 3, ..., are 17, 24, 24, 18, 18, 22, 18, 15, 19, 19, ..., 17, 14, 16, 13, 13, 13, 16, 17, 15, and 13.  A period is considered to start with a negative minima value and end with a positive minima value.  A "peak" occurs when there is a positive spike in minima values (at the end of a period) immediately followed by a negative spike.  The difference in m values between successive peaks in minima values appears to be fairly constant (usually 612, but as small as 528 or as large as 696).  Similar results apply for prime m values of the form 12k-5, 12k-7, or 12k+1.  Let x denote one-twelveth of the difference in m values between successive peaks in minima values.  For prime m values less than 2000000, the x values range from 31 to 107, the distribution of x values for 31, 32, 33, ..., 71 is 2, 5, 3, 11, 14, 22, 29, 29, 46, 98, 110, 156, 171, 228, 394, 418, 563, 697, 1047, 1578, 1561, 1367, 953, 819, 823, 477, 333, 270, 190, 175, 102, 70, 70, 40, 46, 24, 12, 10, 6, 9, and 3 respectively (a population centered around x=51), and the distribution of x values for 91, 92, 93, ..., 107 is 1, 1, 0, 0, 0, 0, 1, 0, 3, 3, 3, 1, 3, 4, 3, 0, and 1 respectively (a second population centered around x=102).  Each of these populations has a Weibull probability distribution.  A Weibull probability plot of  the first population is;



For the first population and prime m values less than 4000000, the shaping parameter is .6692 (with a 95% confidence interval of (.5263, .8510)) and the scaling parameter is 477.4297 (with a 95% confidence interval of (294.3538, 774.3714)).  A Weibull probability plot of the second population of x values for prime m values less than 4000000 is;



For the second population and prime m values less than 6000000, the shaping parameter is 1.1256 (with a 95% confidence interval of (0.8173, 1.5501)) and the scaling parameter is 6.777 (with a 95% confidence interval of (4.6156, 9.9505)).  There should be infinitely many such populations.  The minima appear to be bounded for all m values in a given population.  For example, 2⁴⁰∙1.5704 is an upper bound of the minima (as computed using the m-cycle inequality) in the first population when m is less than 1000000 and is still an upper bound of the minima in the first population when m is less than 6000000.  When c=-1, 2³⁹∙1.2002 is an upper bound of the minima in the first population when m is less than 3000000 and is still an upper bound of the minima in the first population when m is less than 6000000.

The minima in the second population do not increase monotonically within a period; there is a characteristic dip about half-way through the period.  For example, a graph of truncated upper bounds of minima (scaled by 32768.0) for m=609757, 609781, 609877, ..., 610993 (primes of the form 12k+1) is;



Similarly, there is a dip about a third of the way through the period and another dip about two-thirds of the way through the period for the minima in the third population.  For example, a graph of truncated upper bounds of minima (scaled by 32768.0) for m=5905433, 5905553, 5905673, ..., 5907353 (primes of the form 12k+5) is;



A graph of the maximum upper bounds of minima (scaled by 65536.0) for x values from 31 to 72 (in the first population) for prime m values less than 6000000 is;


(The scaled maximum upper bounds of minima for x=31, 32, 33, ..., 52 are 174, 188, 199, 226, 234, 248, 274, 295, 320, 344, 379, 418, 480, 532, 634, 746, 934, 1202, 1760, 3356, 24559, and 26346062 respectively.  This appears to be an exponential probability distribution where μ=1.1993e+06 with a 95% confidence interval of (.8219e+06, 1.9136e+06).  The scaled maximum upper bounds of minima for x=53, 54, 55, ..., 72 appear to be random samples from the same exponential probability distribution.)  A graph of the maximum upper bounds of minima (scaled by 65536.0) for x values from 89 to 114 (in the second population) for prime m values less than 6000000 is;


A graph of the sorted maximum upper bounds of minima for x values from 89 to 144 for prime m values less than 6000000 is;


The periods for prime l values of the form 54k+1 are approximately equal.  Similar results apply for prime l values of the form 54k+5, 54k+7, 54k+11, 54k+13, 54k+17, 54k+19, 54k+23, 54k+25, 54k+29, 54k+31, 54k+35, 54k+37, 54k+41, 54k+43, 54k+47, 54k+49, or 54k+53.  Let x denote one fifty-fourth of the difference in l values between successive peaks in minima values.  For prime l values less than or equal to 3169919, the distribution of x values for 11, 12, 13, ..., 17 is 504, 1488, 2848, 4712, 4054, 1907, and 1009 respectively (a population centered around x=14), the distribution of x values for 25, 26, 27, ..., 32 is 183, 937, 1510, 2826, 2409, 2020, 811, and 165 respectively (a population centered around x=28), the distribution of x values for 40, 41, 42, ..., 46 is 433, 664,1312, 1258, 899, 705, and 184 respectively (a population centered around x=42), etc.  These are Weibull probability distributions (with different parameters).  (The shaping parameters range from about 1.3 to 2.2, so these are "aging" processes.)  For prime l values less than or equal to 6339847, the scaled (by 65536.0) maximum upper bounds of minima (as computed using the m-cycle inequality) for x values of 11, 12, 13, ..., 17 are 6220270, 436632, 744405, 514414, 597, 256, and 166 respectively.  For prime l values less than or equal to 6339847, the scaled maximum upper bounds of minima for x values of 25, 26, 27, ..., 32 are 332686, 1366582, 93783, 526748, 831, 299, 179, and 130 respectively.  For prime l values less than or equal to 6339847, the scaled maximum upper bounds of minima for x values of 40, 41, 42, ..., 46 are 301174, 540827, 101914, 1552, 362, 196, and 139 respectively.  These appear to be exponential probability distributions.  For prime l values less than or equal to 9509761, the scaled maximum upper bounds of minima for x values of 11, 12, 13, ..., 17 are 6220270, 436632, 744405, 916817, 900, 387, and 246 respectively.  For prime l values less than or equal to 9509761, the scaled maximum upper bounds of minima for x values of 25, 26, 27, ..., 32 are 9037278, 1366582, 229310, 526748, 1320, 439, 269, and 195 respectively.  For prime l values less than or equal to 9509761, the scaled maximum upper bounds of minima for x values of 40, 41, 42, ..., 46 are 301174, 540827, 149513, 2343, 521, 294, and 204 respectively.  In general, the scaled maximum upper bounds of minima for the larger x values of each population are still growing.  (As the prime gaps become larger, the scaled maximum upper bounds of minima for even the larger x values of say the first population should stop growing.)

A graph of minima (along the y axis) for m=11, 59, 71, ..., 5099 (primes of the form 12k-1 where m+2 is also a prime) is;


A graph of minima for m=13, 61, 73, ..., 5101 (primes of the form 12k+1 where m-2 is also a prime) is;
 

A graph of minima for m=5, 17, 29, ..., 5021 (primes of the form 12k-7 where m+2 is also a prime) is;


A graph of minima for m=7, 19, 31, ..., 5023 (primes of the form 12k-5 where m-2 is also a prime) is;


A graph of the upper bounds of the minima for m=11, 59, 71, ..., 9767 (primes of the form 12k-1 where m+2 is also a prime) is;



A graph of the upper bounds of the minima for m=5, 17, 29, ..., 9929 (primes of the form 12k-7 where m+2 is also a prime) is;



The x values corresponding to these m values also have a Weibull probability distribution (but with different shaping and scaling parameters).

Tumbles and Jumps

In this section, c is restricted to being 1.  Let y be an even natural number such that y/2 is odd and denote [(y/2)(3/4)^f]+1 where f is a natural number and the brackets denote the greatest integer function by v₁.  (The 3/4 factor corresponds to the "loss" due to a 2 in a sequence vector.)  v₁ will be referred to as a "tumble" if it is odd (tumbles are the counterparts of jumps).  Usually, v₁ is not in the 3n+1 sequence starting with y/2.  Under the same order constraints as least-residue trees, a succession of tumbles and jumps (where the jumps start from the tumbles and the tumbles start from the jumps) have properties similar to a live limb in S of  a least-residue tree.  Denote the peaks and valleys in a succession of tumbles and jumps by y/2, v₁, p₁, v₂, p₂, v₃, p₃, ..., v_i, p_i and denote the exponents of the tumbles by f₁, f₂, f₃, ..., f_i and the exponents of the jumps by g₁, g₂, g₃, ..., g_i.  The conditions to be imposed on y and the jumps are that y be less than the order (of the form 3∙2^k) and greater than  order/2, each of p₁, p₂, p₃, ..., p_i be less than the order, and p_i be greater than order/3.  These conditions do not always guarantee that v₁ is in the 3n+1 sequence starting with y/2, v₂ is in the 3n+1 sequence starting with p₁, v₃ is in the 3n+1 sequence starting with p₂, etc., but they come close to doing so.   For example, for i=2, f₁=5, g₁=4, f₂=5, g₂=3, y=282825586, and an order of 402653184, v₁=33557919, p₁=169886969, v₂=40314975, and p₂=136063043 where y, p₁, and p₂ are within the specified bounds.  The 3n+1 sequence starting with 141412793 (y/2) and having a length of 3f₁+1 ends in 33557920, which is one larger than v₁ (the 3 factor of the tumble exponent corresponds to the number of elements in the 3n+1 sequence due to a 2 in a sequence vector) .  The 3n+1 sequence starting with 169886969 (p₁) and having a length of 3f₂+1 ends in 241889848, where [241889848/6]+1 equals v₂.  Denote the natural numbers these 3n+1 sequences end in by j and k.  Some possibilities for different y values giving tumbles and jumps with exponents of 5, 4, 5, and 3 respectively are; (1) v₁=j and v₂=k, (2) v₁=j-1 and v₂=[k/6]+1, (3) v₁=j-1 and v₂=[k/6], (4) v₁=j and v₂=[k/6], (5) v₁=[j/6]+1 and v₂=6k-5, (6) v₁=j-1 and v₂=k-1, (7) v₁=6j-5 and v₂=k, (8) v₁=[j/6]+1 and 3v₂+1=[k/12], (9) v₁=[j/6]+1 and 3v₂+1=[k/12]+1, and (10) v₁=j-1 and v₂=36k-9 (there are other possibilities).   In general, if  v₁ is in the 3n+1 sequence starting with y/2, there are f₁+1 odd natural numbers and 2f₁ even natural numbers in this sub-sequence (the same as for a sequence vector consisting of f₁ 2's), if v₂ is in the 3n+1 sequence starting with p₁, there are f₂+1 odd natural numbers and 2f₂ even natural numbers in this sub-sequence (the same as for a sequence vector consisting of f₂ 2's), if v₃ is in the 3n+1 sequence starting with p₂, there are f₃+1 odd natural numbers and 2f₃ even natural numbers in this sub-sequence (the same as for a sequence vector consisting of f₃ 2's), etc.  These limbs (not necessarily 1-2 sequence vectors) then have the same X and Y values as a live limb in S of a least-residue tree. 

References

(1)  R. E. Crandall, On the "3x+1" Problem, Mathematics of Computation, Vol. 32, No. 144, Oct. 1978, Pgs. 1281-1291.

(2)  John H. Conway, Unpredictable Iterations, Proc. 1972 Number Theory Conference, University of Colorado, Boulder, CO. 1972, pp. 49-52.

(3)  Terras, R., A stopping time problem on the positive integers, Acta Arithmetica 30 (1976), 241-252.

(4)  Bohm, C., and Sontacchi, G., On the existence of cycles of given length in integer sequences like x_n+1=x_n/2 if x_n even, and x_n+1=3x_n+1 otherwise, Atti Accad. Naz. Lincei, VIII Ser., Rend., Cl. Sci. Fis. Mat. Nat. LXIV (1978), 260-264.

(5)  G. J. Wirsching, The Dynamical System Generated by the 3n+1 Function, 1681, Springer-Verlag (1998).

(6)  R. P. Steiner, A theorem on the Syracuse problem, Proceedings of the 7th Manitoba Conference on Numerical Mathematics and Computation, 1977, pp. 553-559.
 
(7)  L. Halbeisen and N. Hungerbühler, Optimal bounds for the length of rational Collatz cycles, Acta Arith., LXXVIII.3, (1997), pgs. 227-239.

(8)  J. L. Simons, On the Nonexistence of 2-cycles for the 3n+1 Problem, Mathematics of Computation, Dec. 8, 2004, Vol. 74, No. 251, pgs. 1565-1572.

(9)  J. Simons and B. de Weger, Theoretical and computational bounds for m-cycles of the 3n+1 problem, Acta Arith. 2005.

(10)  Edward G. Belaga [2003]: 21.  "Effective polynomial upper bounds to perigees and numbers of (3x + d)-cycles of a given Oddlength", Acta Arithmetica 106, 197206.

(11)  Georges Rhin, "Approximants de Padé et mesures effectives d'irrationalité", Progress in Mathematics 71 [1987], 155-164.

(12)  Hardy, G. H. and Littlewood, J. E., Some Problems of Diophantine Approximation, Acta Math. 37 (1914), 193-239.

(13)  Edward G. Belaga, Maurice Mignotte [2000]:  "Cyclic Structure of Dynamical Systems Associated with 3x+d Extensions of Collatz Problem".  U. Strasbourg report 2000-18, 57 pages.  http://hal.archives-ouvertes.fr/IRMA-ACF, file hal-00129656.

(14)  Mihăilescu, P., Primary Cyclotomic Units and a Proof of Catalan's Conjecture, J. Reine angew. Math. 572: 167-195.

Software

MSVC++™ C programs were used to confirm the above propositions.

Use test0a to find cycles in the 3n+c sequence.  The "iters" variable specifies the number of jumps from the initial n value in the sequence (an odd natural number divisible by 3).  Usually, "iters" is set to 1.  The list of cycles for c in the range from -151 to 151 was generated using this setting.  The list of cycles was confirmed to be the same using an "iters" setting of 2 or 3.  The cycles were verified to be the same as those given by Keith Matthews' cycle-finding program at "http://www.numbertheory.org/keith.html".

Use test0b to regenerate a cycle in the 3n+c sequence given an entry point (must be even) into the cycle.  The order of the cycle is computed.  The output of "test0a" is input to this program.

Use test0c to regenerate cycles in the 3n+c sequence given a list of entry points into the cycles.  (The list of cycles is confirmed.)

Use test0d to regenerate cycles in the 3n+c sequence given a list of entry points into the cycles.  A check for 1-2 sequence vectors at entry points into the cycles is made (for "multiple jump" connection points).

Use test0e to regenerate cycles in the 3n+c sequence given a list of entry points into the cycles.  A check for 1-2 sequence vectors at entry points into the cycles is made (for "jumps over" connection points).

list consists of the cycles found for c in the range from -151 to 151.

list1 consists of one-attachment-point cycles found for c in the range from -151 to 151.

Use test1 to find the number of jumps in the 3n+1 sequence before an even natural number is reached (starting with an odd natural number divisible by 3).

Use test2 to generate a histogram of limb lengths for a 3n+c sequence that is not an "odd" path.  The fourth-to-last element of the sequence must be an odd natural number greater than order/6 and the third-to-last element of the sequence must be an even natural number divisible by 8.  Use test2a to compute the (l, m) values of generalized dead limbs.  The "iters" variable is the number of primary attachment points (usually, the "iters1" variable is set to the same value).

Use test3a to generate limbs of least-residue trees where at least one element of the limb is divisible by 8.

Use test3b to generate limbs of least-residue trees where at least one element of the limb is divisible by 8.  Multiple-word arithmetic is used.

Use test3c to generate limbs of least-residue trees where at least one element of the limb is divisible by 8.  Multiple-word arithmetic is used.  This C program is for use on the TMS320C64™ digital signal processing chip (with hand-optimized assembly language subroutines).

Use test4a12, test4a13, test4a14, test4a15, test4a16, test4a17, test4a18, test4a19, test4a20, test4a21, test4a22, test4a23, test4a24, test4a25, test4a26, test4a27, test4a28, test4a29, test4a30, test4a31, test4a32, and test4a33 to compare the number of limbs in S of a given length to the estimated number of limbs.  Only live limbs are considered.

Use test5a to generate tumbles and jumps having three peaks and two troughs.

Use test5b to generate tumbles and jumps having four peaks and three troughs.

Use test6a to compute the largest y values, the smallest y values, the largest z values, and the smallest z values for live limbs in S associated with a given sequence vector.  The maximum |X(δ/y)| value is computed.

Use test6b to compute the largest y values for live limbs in S associated with a given sequence vector.  The maximum |X(δ/y)| value is computed.  This C program is for use on the TMS320C64™ digital signal processing chip (with hand-optimized assembly language subroutines).

Use test7 to check that n(order/2)>ce, c>0, n>0, or (-n)(order/2)>(-c)e, n<0, c<0.  Floating point arithmetic is used.

Use test8 to check that x>0 for limb lengths with positive limits and x<0 for limb lengths with negative limits.

Use test9 to generate least-residue trees.  The permutation of A, B, C, and D for different c values is checked.

Use test10 to generate least-residue trees for the 3n-1 sequence, k≤15.  Numerous properties of least-residue trees are tested.  Limbs in S having lengths of 5, 10, 15, 18, 23, and 28 are verified to be dead.

Use test11 to generate long limbs in S.  Sequence vectors are generated.

Use test12 to find sequence vectors corresponding to different e values.  All the orders up to 3∙2²⁴ are computed.  The maximum and minimum x/y values for live limbs are computed for every order and are verified to be either ascending of descending.

Use test13 to compute a and b values.

Use test14a to compute the minimum element in a cycle using the formula M_l,m=∑(]jm/l[-](j-1)m/l[)2^j-13^m-]jm/l[  where the summation is from j=1 to l.  Use test14b to compute the minimum element in a cycle in a limb of S of a least-residue tree.  Use test14c to compute the minimum element in a cycle in a limb of S of a least-residue tree (using the TMS320C64™ DSP.)  Use test14d to compute the minimum element in a cycle in a limb of S of a least-residue tree when m and m+2 are primes.  Use test14e to output the computed minima for a cycle in a limb of S of a least-residue tree.  Use test14f to compute an upper bound of the minimum element in a cycle in a limb of S of a least-residue tree where m and m+2 are prime.  Use test14g to compute an upper bound of the minimum element in a cycle in a limb of S of a least-residue tree when m is prime.  Use test14h to compute an upper bound of the minimum element in a cycle in a limb of S of a least-residue tree when m is prime (using the TMS320C64™ DSP).  Use test14i to generate the sequence vector corresponding to the smallest e from the parity vector.  Use test14j to compute the largest odd element in a cycle using the floor function.  Use test14k to compute an upper bound of the minimum element in a cycle in a limb of S of a least-residue tree when l is prime (using the TMS320C64™ DSP).

Use test15 to generate parity vectors (distinct under rotation) for given l and m values (using the TMS320C64™ DSP).  Subroutines called are bitcnt and pack.

Use test16 to determine that the (l, m) value of a generalized dead limb minus the (l, m) value of a generalized continued-fraction convergent of log(3)/log(2) is the (l, m) value of a shorter generalized dead limb.

Use test17 to show that |2^l-3^m| increases monotonically when (l, m) are generalized continued-fraction convergents of log(3)/log(2) (excluding (2, 1), (4, 2), (6, 4), and (9, 6)).  |2^l-3^m| is shown to increase monotonically for 105 generalized continued-fraction convergents of log(3)/log(2).  This program is for use on the TMS320C64™ DSP.

Use test18 to generate the parity vector of a 3n+c cycle where twice the minimum element in the cycle is larger than the maximum odd element in the cycle (and the elements are large).  This program is for use on the TMS320C64™ DSP.  Use test18a to compute M_l,m/(2^l-3^m) values for the corresponding M-cycles.

Use test19 to find rotations of the parity vector ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l, where (0, 1, 0) is at the beginning of the vector and twice the smallest s value is larger than the largest "odd" s value.  This program is for use on the TMS320C64™ DSP.  Use test19a to find rotations of the vector satisfying the two additional conditions.

Use test20 to find limbs in S almost having the parity vector ]jm/l[ - ](j-1)m/l[, j=1, 2, 3, ..., l (the parity vector is rotated so that a (1, 1) is at the beginning of the vector and the first 1 in the vector is changed to a 0 and the last element of the vector [a 0] is changed to a 1).  A subroutine used is limb.  This program is for use on the TMS320C64™ DSP.  test20a doesn't use the "limb" subroutine.

Use test21 to compute the number of odd elements in a one-jump (or multiple-jump) attachment point minus j where t≡u(mod 2^j).  test21a computes these values for c values between 601 and 997.  test21b computes these values for c values between 1001 and 1499.  test21c computes these values for c values between 1501 and 1999.  test21d computes these values for c values between 2003 and 2497.  The properties of no-jump, one-jump, multiple-jump, and jumped-over attachment points are tested.

Use log2 to compute the logarithm of 2.  Use log32 to compute the logarithm of 3/2.  Use expand to compute the generalized continued-fraction convergents of log(3)/log(2).  Use test22 to compute the upper bound of the minimum in a 3n+c cycle using Proposition (33).

Multiple-word arithmetic C subroutines are add64, add128, add256, add512, add1024, carry, copy256, copy512, copy1024, div6432, div12864, div25632,  lshift256, lshift512, lshift1024, lmbd, mul6432, mul6464, mul12832, set256, set512, set1024, shift64, shift128, shift256, shift512, shift1024, sub64, sub128, sub256, sub512, sub1024, subr2, n-word arithmetic, table, table1, and table2.

Multiple-word arithmetic TMS320C64™ assembly language subroutines are add64, div12864, div6432, mul6432, mul6464, mul3232, shift64, sub64, subr2, and n-word arithmetic.