Prime-value pattern in Erdős Problem 1141

Are there only finitely many integers n for which n−k² is prime whenever k≥1, k²<n and gcd(k,n)=1? The coprimality condition is essential to the original a=1 formulation of Erdős Problem 1141.

Why it matters and what counts as progress

Why it matters: The finiteness claim rules out an infinite family in which every admissible square subtraction remains prime.

What counts as progress: Prove finiteness under the full quantifier on k or obtain explicit bounds on the possible exceptional n.

How the problem developed

1999 · Open question [1][2]
Paul Vardi Asked whether 968 is the largest n for which n−k² is prime for every admissible k; the cited example 968 contains an error, but the underlying infinitude question remained open.
8 Apr 2026 · Finiteness theorem [2][1]
An internal OpenAI model; manuscript by Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke and Gregory Valiant Proved that for each integer a≥1 only finitely many n have n−ak² prime for every admissible k, which rules out the proposed infinitude when a=1.