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Claim. There are at most nine positive integers $n$ such that $n-2x^2$ is prime for every integer $x$ with $2x^2<n$: the eight known ones, $2,5,7,13,31,61,181,199$, and possibly one more. The answer to Problem 1140 is therefore no.

Source. M. Epure and A. Gica, Principal quadratic real fields in connection with some additive problems, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53(101) (2010), no. 3, 251–259; the paper prints received 2009-07-04 and revised 2010-03-20, and the record gives no publication date finer than the year 2010, by which this page is named. The supporting result is R. A. Mollin and H. C. Williams, Period four and real quadratic fields of class number one, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 4, 89–93, doi:10.3792/pjaa.65.89.

The argument. An even $n$ has $n-0$ prime only for $n=2$. If $n$ is odd and $n-2x^2$ is prime for every $x$ with $2x^2<n$, then $m=2n$ satisfies: $m-a^2$ is twice a prime for every even $a\ge0$ with $a^2\le m$ (the paper's own condition is $2x^2\le n$, which differs from the site's strict inequality only at $n=2$). For odd $n$ the residue of $m$ modulo $8$ splits the problem.

• $n\equiv1\pmod 4$, so $m\equiv2\pmod 8$. Theorem 4.1 of the paper shows that such an $m$ is $x^2+1$ with $x$ prime, that $\mathbb{Z}[\sqrt m]$ has class number two, and, through Byeon and Lee's determination of the odd $x$ with $h(x^2+1)=2$ (earlier Mollin and Williams under the generalized Riemann hypothesis), that $m\in\{10,26,122,362\}$. Hence $n\in\{5,13,61,181\}$.
• $n\equiv3\pmod 4$, so $m\equiv6\pmod 8$. Remark 2 of the paper (p. 258) sketches the same path: $m=(4y)^2-2$ with $\mathbb{Z}[\sqrt m]$ principal, and the class-number-one result of Mollin and Williams for this family gives $m\in\{14,62,398\}$ with at most one further exception, whose existence is not settled. Hence $n\in\{7,31,199\}$ and at most one more.

The paper states the connection to this problem itself: after listing $5,7,13,31,61,181,199$ it concludes that "besides these numbers it could only exist one more number N with the afore-mentioned property" (Remark 2, p. 258). The $n\equiv3\pmod 4$ case is a remark with a sketched argument, not a numbered theorem with a written proof; the finiteness conclusion is what the site accepts.

Acceptance. Refereed: Bull. Math. Soc. Sci. Math. Roumanie 53(101), no. 3 (2010). Reviewed: the site's curator, Thomas Bloom, labels the problem disproved and credits Theorem 4.1 of Epure and Gica for the residue class $1\pmod 4$ and their remark with the Mollin–Williams result for the class $3\pmod 4$ (problem page last edited 2026-01-26; the community database records the disproved status from 2026-01-24). A forum comment of 2026-01-24 brought the paper to the thread. No Lean formalization of the result is recorded, and this corpus has not independently verified the proof.

Depends on. No page of this wiki; the result rests on the two cited papers.