Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 1058

claims/: The 1 claim page of Problem 1058, one per claimant's result; the problem's standing derives from them.

Statement. 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 $p_{k+1}$ ?

Status. Proved.

Source. erdosproblems.com/1058, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1058, https://www.erdosproblems.com/1058.

References.

[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_unsolved_problems_number_theory.
[Lu01] Luca, Florian, On a conjecture of Erdős and Stewart. Math. Comp. 70 (2001), no. 234, 893-896.

Formalization. The statement declaration in formal-conjectures (pinned at the commit of 2026-09-20 that added the file) is closed by sorry, but it is tagged as solved and carries a formal-proof attribute pointing at a Lean 4 proof in Boris Alexeev's repository (src/latest/ErdosProblems/Erdos1058.lean, pinned to the commit the claim page links), which declares itself a formalization of Luca's solution and proves that the solutions are exactly $n=1,2,3,4,5$ . This corpus has not built or audited that proof, so it is recorded as the claimant's formalization link on the claim page and as no formalized evidence.

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Linked library material

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Linked from (8)

Luca proves the Erdős–Stewart conjecture problems/diophantine_problems/E1058/claims Diophantine Problems and Powers Diophantine Problems and Powers diophantine_problems/luca_2001_conjecture_erdos_stewart Lemma (p. 893): a solution of n! + 1 = p_k^a p_(k+1)^b with n >= 12 has ab != 0 Theorem (p. 893): n! + 1 = p_k^a p_(k+1)^b with p_(k-1) <= n < p_k has no solution with n >= 6 number_theory/guy_2004_unsolved_problems_number_theory

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