Problem 779

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Statement. Let $n> 1$ and $p_1<\cdots<p_n$ denote the first $n$ primes. Let $P=\prod_{1\leq i\leq n}p_i$. Does there always exist some prime $p$ with $p_n<p<P$ such that $P+p$ is prime?

Status. Falsifiable: the site's label, an open problem that a single finite counterexample would disprove.

Source. erdosproblems.com/779, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #779, https://www.erdosproblems.com/779.

Formalization. Statement in formal-conjectures.

Current assessment

No literature search or independent assessment is recorded on this page; the Status sentence gives the site's label. That label, Falsifiable, records an open question whose negation one finite counterexample would witness: a counterexample is a single $n>1$ for which no prime $p$ with $p_n<p<P$ makes $P+p$ prime, a check by finite arithmetic over the primes below $P$, while a proof must cover every $n$. The label is a body note, not a claim, and the problem has no claim page. The site's commentary attributes the question to Deaconescu, records his verification of it for $n\le1000$, and reports Erdős's expectation that the least such $p$ is at most a fixed power of $n$; its probabilistic heuristic makes a failure at any $n$ extremely unlikely. Nothing here is independently reviewed.