Is the Syracuse falling time bounded by 12?
Résumé
Let T : N → N denote the 3x + 1 function, defined by T (n) = n/2 if n is even, T (n) = (3n + 1)/2 if n is odd. As an accelerated version of T , we define the jump function jp(n) = T^(l) (n) for n ≥ 1, where l = log 2 (n) + 1 = the number of digits of n in base 2. Correspondingly, we define the falling time ft(n) for n ≥ 2 as the least k ≥ 1, if any, such that jp^(k) (n) < n, or ∞ otherwise. The Collatz conjecture is equivalent to ft(n) < ∞ for all n ≥ 2. As a stronger form of it, is it conceivable that ft(n) be uniformly bounded? So far, the highest value of ft(n) we have found is 15. The related Syracuse function is defined on odd integers n ≥ 1 by syr(n) = (3n+1)/2 ν , the largest odd factor of 3n+1. Correspondingly, we define the Syracuse jump function sjp(n) = syr^(l) (n), where again l = log 2 (n) + 1, and the Syracuse falling time sft(n) for odd n ≥ 3, namely the least k ≥ 1 such that sjp^(k) (n) < n, or ∞ otherwise. Again, the Collatz conjecture is equivalent to sft(n) < ∞ for all odd n ≥ 3. So far, the highest value of sft(n) we have found is 10, including among all known glide records and all n = 2^l − 1 for l ≥ 2. Based on the computational and heuristic evidence presented here, we conjecture that both ft(n) and sft(n) are uniformly bounded.
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