On the nonexistence of 2-cycles for the 3𝑥+1 problem

Simons, John

doi:10.1090/S0025-5718-04-01728-4

On the nonexistence of $2$-cycles for the $3x+1$ problem
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by John L. Simons;

Math. Comp. 74 (2005), 1565-1572

DOI: https://doi.org/10.1090/S0025-5718-04-01728-4

Published electronically: December 8, 2004

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Abstract:

This article generalizes a proof of Steiner for the nonexistence of $1$-cycles for the $3x+1$ problem to a proof for the nonexistence of $2$-cycles. A lower bound for the cycle length is derived by approximating the ratio between numbers in a cycle. An upper bound is found by applying a result of Laurent, Mignotte, and Nesterenko on linear forms in logarithms. Finally numerical calculation of convergents of $\log _2 3$ shows that $2$-cycles cannot exist.

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