Problem 1140

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claims/: The 1 claim page of Problem 1140, one per claimant's result; the problem's standing derives from them.

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Statement. Do there exist infinitely many $n$ such that $n-2x^2$ is prime for all $x$ with $2x^2<n$?

Status. Disproved: the site credits Epure and Gica's Theorem 4.1 and their remark with a result of Mollin and Williams, which together leave at most nine such $n$; the accepted claim page is Epure and Gica 2010.

Source. erdosproblems.com/1140, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1140, https://www.erdosproblems.com/1140.

References.

• [EpGi10] Epure, Mihai and Gica, Alexandru, Principal quadratic real fields in connection with some additive problems. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) (2010), 251-259.
• [MoWi89] Mollin, R. A. and Williams, H. C., Period four and real quadratic fields of class number one. Proc. Japan Acad. Ser. A Math. Sci. (1989), 89-93.

Formalization. None recorded.

Current assessment

The question, in the site's formulation accessed, asks for infinitely many $n$ with $n-2x^2$ prime for every $x$ with $2x^2<n$. The answer is no: at most nine such $n$ exist, the eight known ones $2,5,7,13,31,61,181,199$ and possibly one more. Doubling $n$ turns the condition into one on $m=2n$ that Epure and Gica study, and the residue of $n$ modulo $4$ splits the cases: Theorem 4.1 of Epure and Gica 2010 gives exactly $5,13,61,181$ for $n\equiv1\pmod 4$, and their Remark 2, with the class-number-one result of Mollin and Williams [MoWi89], gives $7,31,199$ and at most one further exception for $n\equiv3\pmod 4$. The possible exception is the possible further field of class number one in the Mollin–Williams family, whose existence neither paper settles; it does not affect the finiteness.

Acceptance rests on the refereed paper and on the site's curator labeling the problem disproved with that credit; this corpus has not verified the proof, and the $n\equiv3\pmod 4$ case is argued in a remark rather than a numbered theorem. A conditional argument posted in the forum thread on 2026-01-25, assuming the generalized Riemann hypothesis, was found there to have a gap and is superseded by the unconditional result; it has no claim page. Search scope: the site's problem page and forum thread, the community database, and the two cited papers (2026-10-07). No Lean formalization of the result is known.