Erdős Problem #1058

Let $2=p_1<p_2<\cdots$ be the sequence of prime numbers. Are there only finitely many $n$ such that $n\in [p_{k-1},p_k)$ and the only primes dividing $n!+1$ are $p_{k}$ and $p_{k+1}$?

#1058: [Gu04]

number theory

A conjecture of Erdős and Stewart, as reported in problem A2 of Guy's collection [Gu04]. The only known cases are $n=1,2,3,4,5$.

Luca [Lu01] proved that indeed these are the only solutions.

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This page was last edited 28 September 2025.

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Additional thanks to: Terence Tao

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