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

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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 $v < r$.

- Robert Baillie, G. Cormack, and H. C. Williams,
*The problem of Sierpiński concerning $k\cdot 2^{n}+1$*, Math. Comp.**37**(1981), no. 155, 229–231. MR**616376**, DOI https://doi.org/10.1090/S0025-5718-1981-0616376-2 - N. S. Mendelsohn,
*The equation $\phi (x)=k$*, Math. Mag.**49**(1976), no. 1, 37–39. MR**396385**, DOI https://doi.org/10.2307/2689883 - Oystein Ore, J. L. Selfridge, and P. T. Bateman,
*Advanced Problems and Solutions: Solutions: 4995*, Amer. Math. Monthly**70**(1963), no. 1, 101–102. MR**1532000**, DOI https://doi.org/10.2307/2312814 - W. Sierpiński,
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*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.

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Article copyright:
© Copyright 1983
American Mathematical Society