A noniterative $2$-adic statement of the $3N+1$ conjecture
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- by Daniel J. Bernstein
- Proc. Amer. Math. Soc. 121 (1994), 405-408
- DOI: https://doi.org/10.1090/S0002-9939-1994-1186982-9
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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.References
- Ethan Akin, $3x + 1$, unpublished manuscript.
- R. E. Crandall, On the $“3x+1''$ problem, Math. Comp. 32 (1978), no. 144, 1281–1292. MR 480321, DOI 10.1090/S0025-5718-1978-0480321-3
- Donald E. Knuth, The art of computer programming. Vol. 2, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms. MR 633878
- Jeffrey C. Lagarias, The $3x+1$ problem and its generalizations, Amer. Math. Monthly 92 (1985), no. 1, 3–23. MR 777565, DOI 10.2307/2322189
- Helmut Müller, Das “$3n+1$”-Problem, Mitt. Math. Ges. Hamburg 12 (1991), no. 2, 231–251 (German). Mathematische Wissenschaften gestern und heute. 300 Jahre Mathematische Gesellschaft in Hamburg, Teil 2. MR 1144786
- Ray Steiner, On the “$QX+1$ problem”, $Q$ odd, Fibonacci Quart. 19 (1981), no. 3, 285–288. MR 627405
Bibliographic Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 405-408
- MSC: Primary 11S85; Secondary 11B75
- DOI: https://doi.org/10.1090/S0002-9939-1994-1186982-9
- MathSciNet review: 1186982