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 $p_{k+1}$ ?

Source: erdosproblems.com/1058

Number theory

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An accepted solution exists. The statement is true.

Gu04: Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; section A2 "Primes connected with factorials", printed p. 11, which states the conjecture of Erdős and Stewart that $n=1,\ldots,5$ are the only cases with $n!+1=p_k^ap_{k+1}^b$ and $p_{k-1}\le n<p_k$ , and records the 1998 announcement of its proof by Flammenkamp and Luca; the proof is [Lu01]. Library home: Guy 2004.
Lu01: Luca, Florian, On a conjecture of Erdős and Stewart. Math. Comp. 70 (2001), no. 234, 893-896.

Problem 1058

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