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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A noniterative $ 2$-adic statement of the $ 3N+1$ conjecture


Author: Daniel J. Bernstein
Journal: Proc. Amer. Math. Soc. 121 (1994), 405-408
MSC: Primary 11S85; Secondary 11B75
MathSciNet review: 1186982
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Abstract: Associated with the $ 3N + 1$ problem is a permutation $ \Phi $ of the 2-adic integers. The $ 3N + 1$ conjecture is equivalent to the conjecture that 3Q is an integer if $ \Phi (Q)$ is a positive integer. We state a new definition of $ \Phi $. To wit: Q and $ N = \Phi (Q)$ are linked by the equations $ Q = {2^{{d_0}}} + {2^{{d_1}}} + \cdots $ and $ N = ( - 1/3){2^{{d_0}}} + ( - 1/9){2^{{d_1}}} + ( - 1/27){2^{{d_2}}} + \cdots $ with $ 0 \leq {d_0} < {d_1} < \cdots $. We list four applications of this definition.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1186982-9
PII: S 0002-9939(1994)1186982-9
Article copyright: © Copyright 1994 American Mathematical Society