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The problem of Sierpiński concerning $ k\cdot 2\sp{n}+1$


Authors: Robert Baillie, G. Cormack and H. C. Williams
Journal: Math. Comp. 37 (1981), 229-231
MSC: Primary 10A25
DOI: https://doi.org/10.1090/S0025-5718-1981-0616376-2
Corrigendum: Math. Comp. 39 (1982), 308.
Corrigendum: Math. Comp. 39 (1982), 308.
MathSciNet review: 616376
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Abstract: Let $ {k_0}$ be the least odd value of k such that $ k \cdot {2^n} + 1$ is composite for all $ n \geqslant 1$. In this note, we present the results of some extensive computations which restrict the value of $ {k_0}$ to one of 119 numbers between 3061 and 78557 inclusive. Some new large primes are also given.


References [Enhancements On Off] (What's this?)

  • [1] G. Cormack & H. C. Williams, "Some very large primes of the form $ k \cdot {2^m} + 1$," Math. Comp., v. 35, 1980, pp. 1419-1421. MR 583519 (81i:10011)
  • [2] P. Erdös & A. M. Odlyzko, "On the density of odd integers of the form $ (p - 1){2^{ - n}}$ and related questions," J. Number Theory, v. 11, 1979, pp. 257-263. MR 535395 (80i:10077)
  • [3] R. K. Guy, "Some unsolved problems," in Computers in Number Theory (A. O. L. Atkin and B. J. Birch, Eds.), Academic Press, New York, 1971, pp. 415-422. MR 0277393 (43:3126)
  • [4] J. L. Selfridge, "Solution to problem 4995," Amer. Math. Monthly, v. 70, 1963, p. 101. MR 1532000
  • [5] W. Sierpiński, "Sur un problème concernant les nombres $ k \cdot {2^n} + 1$," Elem. Math., v. 15, 1960, pp. 73-74; Corrigendum, v. 17, 1962, p. 85. MR 0117201 (22:7983)
  • [6] W. Sierpiński, 250 Problems in Elementary Number Theory, American Elsevier, New York, 1970, p. 10 and p. 64.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1981-0616376-2
Article copyright: © Copyright 1981 American Mathematical Society

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