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On the $ \lq\lq 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: It is an open conjecture that for any positive odd integer m the function

$\displaystyle C(m) = (3m + 1)/{2^{e(m)}},$

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)$.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1978-0480321-3
Keywords: Algorithm, diophantine equation
Article copyright: © Copyright 1978 American Mathematical Society

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