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Updated Oct 7

Problem 454

Statement. Let

f(n) = \min_{i<n} (p_{n+i}+p_{n-i}),

where $p_k$ is the $k$ th prime. Is it true that

\limsup_n (f(n)-2p_n)=\infty?

Formulation. The minimum is taken over $1\le i<n$ . Pomerance [Po79] defines the same quantity over $0<i<n$ and proves $f(n)>2p_n$ for infinitely many $n$ , so the $\limsup$ is at least $2$ , as the site's commentary records; McNew's quantity (23) and the formal-conjectures statement also take $1\le i<n$ . Admitting $i=0$ would give $f(n)\le 2p_n$ for every $n$ and the trivial answer no, so the site's $\min_{i<n}$ is read as the sources read it.

Status. Open.

Source. erdosproblems.com/454, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #454, https://www.erdosproblems.com/454.

References.

[Po79] Pomerance, Carl, The prime number graph. Math. Comp. (1979), 399-408.

Formalization. Statement in formal-conjectures, at the linked commit of 18 September 2026, which defines f n as the infimum over $0<i<n$ and carries no formal_proof annotation; its variant two_le_limsup, the Pomerance bound, is marked research solved with a sorry proof.

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Linked library material

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Linked from (6)

Primes primes/mcnew_2018_convex_hull_prime_number_graph Equation (23) (p. 13): the quantity M_n, with the midpoint convex prime data of Section 5 Theorem 2.2 (p. 5): the convex primes up to x number O(x^{2/3}/log^{2/3} x)Theorem 2.3 (p. 5) with Corollaries 2.5, 2.7, 2.8: gaps between convex primes primes/pomerance_1979_prime_number_graph

Graph