Problem 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?
Categories:
Number Theory Primes
Progress
A problem of Deaconescu. Erdős expects that the least such prime is much smaller than $P$, and in fact satisfies $p\leq n^{O(1)}$. Deaconescu has verified this conjecture for $n\leq 1000$.With the usual heuristic, we expect that $P+p$ is prime with 'probability' $\approx 1/\log P$, and hence the chance that this fails is $\ll (1-1/\log P)^{P-p_n}\ll \exp(-n^{-cn})$, using $P=n^{(1+o(1))n}$. As Cambie points out, 'the chances of failing are ridiculously small'.
Source: erdosproblems.com/779 | Last verified: January 16, 2026