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Journal Article

# On the Smallest $k$ Such that All $k \cdot 2^N + 1$ are Composite

G. Jaeschke
Mathematics of Computation
Vol. 40, No. 161 (Jan., 1983), pp. 381-384
DOI: 10.2307/2007382
Stable URL: http://www.jstor.org/stable/2007382
Page Count: 4

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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 $v < r$.