On the $“3x+1”$ problem

Author:
R. E. Crandall

Journal:
Math. Comp. **32** (1978), 1281-1292

MSC:
Primary 10A99

DOI:
https://doi.org/10.1090/S0025-5718-1978-0480321-3

MathSciNet review:
0480321

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Abstract | References | Similar Articles | Additional Information

Abstract: It is an open conjecture that for any positive odd integer *m* the function \[ C(m) = (3m + 1)/{2^{e(m)}},\] where $e(m)$ is chosen so that $C(m)$ is again an odd integer, satisfies ${C^h}(m) = 1$ for some *h*. Here we show that the number of $m \leqslant x$ which satisfy the conjecture is at least ${x^c}$ for a positive constant *c*. A connection between the validity of the conjecture and the diophantine equation ${2^x} - {3^y} = p$ is established. It is shown that if the conjecture fails due to an occurrence $m = {C^k}(m)$, then *k* is greater than 17985. Finally, an analogous "$qx + r$" problem is settled for certain pairs $(q,r) \ne (3,1)$.

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Keywords:
Algorithm,
diophantine equation

Article copyright:
© Copyright 1978
American Mathematical Society