On the Collatz $3n+1$ algorithm
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- by Lynn E. Garner PDF
- Proc. Amer. Math. Soc. 82 (1981), 19-22 Request permission
Abstract:
The number theoretic function $s(n) = \tfrac {1} {2}n$ if $n$ is even, $s(n) = 3n + 1$ if $n$ is odd, generates for each $n$ a Collatz sequence $\{ {{s^k}(n)} \}_{k = 0}^\infty$, ${s^0}(n) = n$, ${s^k}(n) = s({s^{k - 1}}(n))$. It is shown that if a Collatz sequence enters a cycle other than the $4,2,1,4, \ldots$ cycle, then the cycle must have many thousands of terms.References
- C. J. Everett, Iteration of the number-theoretic function $f(2n)=n,$ $f(2n+1)=3n+2$, Adv. Math. 25 (1977), no. 1, 42–45. MR 457344, DOI 10.1016/0001-8708(77)90087-1 M. Gardner, Mathematical games, Sci. Amer. 226 (1972), 115.
- Riho Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976), no. 3, 241–252. MR 568274, DOI 10.4064/aa-30-3-241-252
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 82 (1981), 19-22
- MSC: Primary 10L10; Secondary 10A35
- DOI: https://doi.org/10.1090/S0002-9939-1981-0603593-2
- MathSciNet review: 603593