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Source. T. Feng, T. Trinh, G. Bingham et al., Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems, arXiv:2601.22401v3 (5 February 2026); Section 4.3, the problem and Remark 4.3 on p. 24, the solution on pp. 24--26. The result is unnumbered. The artifact is identified on the source card.

Read depth. Claims checked: the assertion and the proof (pp. 24--26) were read through on the print; the list of two-distance four-point configurations is asserted in the paper from "elementary observations" (p. 25) and was not checked here, nor was the cited counting theorem. Nothing here is independently reviewed. A preprint.

Statement

Let $\Lambda=\{m(1,0)+k(\tfrac12,\tfrac{\sqrt7}2):m,k\in\mathbb Z\}$, the ring of integers of $\mathbb Q(\sqrt{-7})$ embedded in $\mathbb R^2$, and let $P_n$ be the $n$ points of $\Lambda$ closest to the origin. The paper proves (pp. 24--26) that $P_n$ determines $O(n/\sqrt{\log n})$ distinct distances and that every $4$ distinct points of $\Lambda$ determine at least $3$ distinct distances; the answer to the question is affirmative.

Proof pointer

Squared distances in $\Lambda$ are values of the form $m^2+mk+2k^2$ of discriminant $-7$ up to $O(n)$, and Bernays's extension of the Landau--Ramanujan theorem counts the integers up to $X$ so represented as $\sim CX/\sqrt{\log X}$ (p. 25). Every planar two-distance four-point set is, by the paper's list, similar to one of six configurations; the isosceles trapezoid needs an irrational squared-distance ratio, and the rest contain a square or an equilateral triangle, neither of which fits in $\Lambda$ because $i$ and $\sqrt{-3}$ are not in $\mathbb Q(\sqrt{-7})$ (pp. 25--26).

Dependencies

Bernays (1912), the asymptotic count of integers represented by a positive definite binary quadratic form (cited, not held).

Bears on

• Problem 659: answers the question as posed affirmatively. The paper classifies the case as an independent rediscovery: Remark 4.3 (p. 24) reports essentially the same result in a 2014 blog post of Sheffer, from an argument of Sheffer and Lund that does not treat the trapezoid configuration, and a later full solution by Grayzel.