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On the nonexistence of $2$-cycles for the $3x+1$ problem
2004
Mathematics of Computation
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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.
doi:10.1090/s0025-5718-04-01728-4
fatcat:glt5rgjqe5cn5jad4zazljk6nu