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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the $“3x+1”$ problem

Author: R. E. Crandall
Journal: Math. Comp. 32 (1978), 1281-1292
MSC: Primary 10A99
MathSciNet review: 0480321
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Abstract: It is an open conjecture that for any positive odd integer m the function \[ C(m) = (3m + 1)/{2^{e(m)}},\] 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 $(q,r) \ne (3,1)$.

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  • J. H. Conway, Unpredictable iterations, Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972) Univ. Colorado, Boulder, Colo., 1972, pp. 49–52. MR 0392904
  • I. N. HERSTEIN & I. KAPLANSKY, Matters Mathematical, 1974.
  • 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
  • M. ABRAMOWITZ & I. STEGUN (Editors), Handbook of Mathematical Functions, 9th printing, Dover, New York, 1965. Zentralblatt für Mathematik, Band 233, p. 10041.
  • Joe Roberts, Elementary number theory—a problem oriented approach, MIT Press, Cambridge, Mass.-London, 1977. MR 0498337
  • A. Ya. Khinchin, Continued fractions, The University of Chicago Press, Chicago, Ill.-London, 1964. MR 0161833
  • Pi Mu Epsilon Journal, v. 5, 1972, pp. 338, 463. S. S. PILLAI, J. Indian Math. Soc., v. 19, 1931, pp. 1-11. A. HERSCHEFELD, Bull. Amer. Math. Soc., v. 42, 1936, pp. 231-234.

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Keywords: Algorithm, diophantine equation
Article copyright: © Copyright 1978 American Mathematical Society