Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

On the smallest $ k$ such that all $ k\cdot 2\sp{n}+1$ are composite


Author: G. Jaeschke
Journal: Math. Comp. 40 (1983), 381-384
MSC: Primary 10A25; Secondary 10-04
DOI: https://doi.org/10.1090/S0025-5718-1983-0679453-8
Corrigendum: Math. Comp. 45 (1985), 637.
MathSciNet review: 679453
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we present some computational results which restrict the least odd value of k such that $ k \cdot {2^n} + 1$ is composite for all $ n \geqslant 1$ to one of 91 numbers between 3061 and 78557,inclusive. Further, we give the computational results of a relaxed problem and prove for any positive integer r the existence of infinitely many odd integers k such that $ k\cdot{2^r} + 1$ is prime but $ k\cdot{2^v} + 1$ is not prime for $ v < r$.


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

  • [1] R. Baillie, G. Cormack & H. C. Williams, "The problem of Sierpinski concerning $ k\cdot{2^n} + 1$," Math. Comp., v. 37, 1981, pp. 229-231. Corrigenda, Math. Comp., v. 39, 1982, p. 308. MR 616376 (83a:10006a)
  • [2] N. S. Mendelsohn, "The equation $ \phi (x) = k$," Math. Mag., v. 49, 1976, pp. 37-39. MR 0396385 (53:252)
  • [3] J. L. Selfridge, "Solution to problem 4995," Amer. Math. Monthly, v. 70, 1963, p. 101. MR 1532000
  • [4] W. Sierpinski, "Sur un probleme concernant les nombres $ k\cdot{2^n} + 1$," Elem. Math., v. 15, 1960, pp. 73-74. MR 0117201 (22:7983)
  • [5] R. G. Stanton & H. C. Williams, Further Results on Coverings of the Integers $ 1 + k\cdot{2^n}$ by Primes, Lecture Notes in Math., vol. 884, Combinatorial Mathematics VIII, pp. 107-114, Springer-Verlag, Berlin and New York, 1980.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10A25, 10-04

Retrieve articles in all journals with MSC: 10A25, 10-04


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1983-0679453-8
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society