Problem Statement
Is there an integer $m$ with $(m,6)=1$ such that none of $2^k3^\ell m+1$ are prime, for any $k,\ell\geq 0$?
Categories:
Primes Covering Systems
Progress
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.)
Source: erdosproblems.com/203 | Last verified: January 14, 2026