Positive odd integers $m$ such that none of $2^km+1$ are prime are called Sierpinski numbers - see [1113] for more details.

Erdős and Graham also ask more generally about $p_1^{k_1}\cdots p_r^{k_r}m+1$ for distinct primes $p_i$, or $q_1\cdots q_rm+1$ where the $q_i$ are primes congruent to $1\pmod{4}$. (Dogmachine has noted in the comments this latter question has the trivial answer $m=1$ - perhaps some condition such as $m$ even is meant.)

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This page was last edited 30 December 2025.