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Proceedings of the American Mathematical Society

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On the Collatz $ 3n+1$ algorithm


Author: Lynn E. Garner
Journal: Proc. Amer. Math. Soc. 82 (1981), 19-22
MSC: Primary 10L10; Secondary 10A35
MathSciNet review: 603593
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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 [Enhancements On Off] (What's this?)

  • [1] 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 0457344
  • [2] M. Gardner, Mathematical games, Sci. Amer. 226 (1972), 115.
  • [3] Riho Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976), no. 3, 241–252. MR 0568274

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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0603593-2
Article copyright: © Copyright 1981 American Mathematical Society