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Lower bounds for the total stopping time of 3X+1 iterates

David Applegate·Jeffrey C. Lagarias
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Abstract

The 3X+1 function T(n) is (3n+1)/2 if n is odd and n/2 if n is even. The total stopping time σ_\infty (n) for a positive integer n is the number of iterations of the 3x+1 function to reach 1 starting from n, and is \infty if 1 is never reached. The 3x+1 conjecture states that this function is finite. We show that infinitely many n have a finite total stopping time with σ_\infty(n) > 6.14316 log n. The proof uses a very large computation. It is believed that almost all positive integers have σ_\infty (n) > 6.95212 \log n. The method of the paper should extend to prove infinitely many integers have this property, but it would require a much larger computation.

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David Applegate, Jeffrey C. Lagarias. 2001-03-08. Lower bounds for the total stopping time of 3X+1 iterates. https://arxiv.org/abs/math/0103054

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