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Claim. Michael Beeson, Tiling a triangle into a prime number of congruent triangles, arXiv:2607.23453, posted 26 July 2026, asserts (Theorem 22) that if a triangle $T$ is cut into $N>3$ congruent triangles not similar to $T$, then $N$ is not prime, and concludes (Corollary 23) that for a prime $N$ some triangle can be cut into $N$ congruent triangles exactly when $N=2$, $N=3$ or $N\equiv1\pmod4$; the exceptions are halving an isosceles triangle, cutting an equilateral triangle in three, a 3-tiling of the 30-60-90 triangle, and the reptilings of a right triangle with integer legs $e,f$ into $e^2+f^2$ pieces. So the primes excluded for Problem 634 are exactly the primes $p>3$ with $p\equiv3\pmod4$, $19$ among them. The proof separates the isosceles and equilateral triangles, where it cites earlier work, and disposes of the commensurable-angle case through Laczkovich's theorem that such a tiling of a non-isosceles triangle is a reptiling; in the incommensurable case Laczkovich's classification leaves the group $3\alpha+2\beta=\pi$, settled in Beeson's earlier papers, and the group $\gamma=2\pi/3$, whose four scalene shapes the paper treats one by one with integer side parametrizations and area equations (its Theorems 18 to 21), deriving each count formula itself; its remark to Theorem 21 notes that Zhang had conjectured that formula and proved it when $a\not\equiv b\pmod3$, and gives an independent proof. The paper's Use of AI section says that Claude (Anthropic) was used for proof-checking and copy-editing, that Lemma 14 was proved on 22 July 2026 by Claude Fable, in response to a prompt by Grigore Roșu, and rewritten by hand, that Lemma 16 was discovered and proved by Claude Fable, and that the count formulas of Theorems 18 to 21, already in Zhang's preprint, were rediscovered by Claude without consulting it.

Covers. The exclusion of every prime $p>3$ with $p\equiv3\pmod4$, and with it the list of prime values of $n$; the positive cases $2$, $3$ and the primes $p\equiv1\pmod4$ are the classical constructions. Nothing is claimed about composite $n$, and the characterization asked for by the problem is not claimed.

Depends on. [[problems/discrete_geometry/E0634/claims/1991_08_01_snover_waiveris_williams|Snover, Waiveris and Williams's claim page]], for the rep-tiling theorem and the reptilings that realize the primes $p\equiv1\pmod4$.

Inputs. The paper's isosceles case (Theorem 5(iii)) cites, for the tile with $\gamma=2\pi/3$, Theorem 11 of Beeson's preprint No prime tiling of an isosceles triangle, arXiv:2607.19572, which was withdrawn on 24 September 2026 with the note that its Lemma 9 is not correct as stated and that the theorem has meantime been proved by Bonfioli (claim page); so that step rests on a withdrawn source. The other inputs are Beeson's papers on isosceles triangles (arXiv:1206.1974), on the case $3\alpha+2\beta=\pi$ (arXiv:1206.2229) and on equilateral triangles (arXiv:1812.07014), the solution of Problem 633 (card), the rationality theorem of Beeson and Zhang (arXiv:2604.01314), Laczkovich's papers of 1995 and 2012, and the rep-tiling theorem of Snover, Waiveris and Williams. Bonfioli's manuscript disputes the printed proofs of several theorems of arXiv:1206.2229, which this paper relies on for the group $3\alpha+2\beta=\pi$.

Standing. Claimed. The preprint has no journal record, and the site's label and remarks are unchanged since 30 December 2025, so no outside acceptance is recorded. The same prime classification was claimed on 17 July 2026 by George (claim page) and on 24 July 2026 by Harries (claim page), by different arguments; Harries's second manuscript calls the three manuscripts contemporaneous and credits Beeson's four scalene count formulas with confirming four rows of its own table.