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Claim. Jan Philipp Harries posted a partial proof claim to the proof-claims thread of [[problems/discrete_geometry/E0634/_index|Problem 634]] on 24 July 2026, with a manuscript dated the same day, No triangle can be cut into nineteen congruent triangles: the prime case of Erdős Problem 634, in a code repository linked above. The claim (Theorem 1) is that for every prime $p>3$ with $p\equiv3\pmod4$ no nondegenerate triangle can be tiled by exactly $p$ congruent triangles with pairwise disjoint interiors; in particular no triangle can be cut into nineteen congruent triangles. Together with the classical constructions for $p=2$, $p=3$ and every prime $p\equiv1\pmod4$, this would determine the prime values of $n$ completely: a triangle can be cut into a prime number $p$ of congruent triangles exactly when $p=2$, $p=3$ or $p\equiv1\pmod4$. The proof imports classification theorems of Laczkovich, of Snover, Waiveris and Williams, and of Beeson and Zhang, and reduces a prime tiling to the configurations whose tile has an angle of $2\pi/3$ and commensurable sides, where its Lemma 9, an elementary arithmetic argument, excludes a prime count. A script in the repository checks the algebra of Lemma 9 in exact arithmetic; that script, linked above as code, is a computer check and not a formalization. The manuscript's disclosure says that an initial version of the arithmetic lemma, the proof assembly and the verification script was generated by OpenAI GPT-5.6 Pro on 24 July 2026 and that Claude (Anthropic) was used as a second-model cross-check; the site's tab names GPT-5.6-pro and Fable 5. The claim was posted under the account jphme. The repository was deprecated on 25 July 2026 and the paper moved to the repository jphme/math-problems, where its corrections and any arXiv link are to appear; that location is the second preprint link, pinned at its first commit.

Submission note. Posted to erdosproblems.com as a proof claim by J.P. Harries (account jphme) on 24 July 2026, giving "GPT-5.6-pro, Fable 5" as the AI used:

We prove that for every prime p > 3 with p ≡ 3 (mod 4), no nondegenerate Euclidean triangle can be tiled by exactly p pairwise interior-disjoint congruent triangles. In particular, no triangle can be cut into nineteen congruent triangles. Combined with the classical constructions realizing p = 2, p = 3, and every prime p ≡ 1 (mod 4), this determines the prime tile counts completely: a triangle can be tiled by a prime number p of congruent triangles if and only if p = 2, p = 3, or p ≡ 1 (mod 4). The proof combines recent classification theorems of Beeson, Laczkovich, and Zhang with an elementary arithmetic argument excluding a prime tile count in the remaining configurations, in which the tile has a 2π/3 angle. Several of the structural results we use are at present available only as preprints. Notes: Arxiv submission in progress, Script for Exact-arithmetic (SymPy) checks of the algebra in Lemma 9 can be found at https://github.com/jphme/no19tiling

Covers. The exclusion of every prime $p>3$ with $p\equiv3\pmod4$, and with it the complete list of prime values of $n$. Nothing is claimed about composite $n$ beyond the classical constructions the manuscript cites, 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 constructions for the primes $p\equiv1\pmod4$.

Inputs. The manuscript's inputs are external theorems, none of them carded as it uses them. It cites the solution of Problem 633 (card) only where that paper restates Laczkovich's classification of commensurable angles (its Theorem 3), the rep-tiling theorem of Snover, Waiveris and Williams (Theorem 4), Laczkovich's list of incommensurable shapes (Theorem 5) and the rationality theorem of Beeson and Zhang, arXiv:2604.01314 (Theorem 7). Its Theorems 6 and 8 import Theorems 1 and 11 of Beeson's preprint No prime tiling of an isosceles triangle, arXiv:2607.19572v1, with Theorem 6 proved in Beeson's paper on the case $3\alpha+2\beta=\pi$, arXiv:1206.2229v3. That preprint was withdrawn on 24 September 2026 with the note that its Lemma 9 is not correct as stated and that its theorem has meantime been proved by Bonfioli (claim page), so the isosceles step of this proof rests on a withdrawn source; Bonfioli's manuscript also disputes the printed proofs of several Group 1 theorems of arXiv:1206.2229. The manuscript notes that the withdrawn arXiv:1206.2228 is not used.

Standing. Claimed. The manuscript has no arXiv or journal record. The site's thread shows no comment on the claim and the site's label and remarks are unchanged (page last edited 30 December 2025), so no outside acceptance is recorded. The same prime exclusion, with the additional value $46$, was claimed a week earlier on [[problems/discrete_geometry/E0634/claims/2026_07_17_george|George's claim page]], and two days later by Beeson (claim page); the three write-ups are by different claimants and are recorded separately. Harries's later manuscript on the composite values has its own page, [[problems/discrete_geometry/E0634/claims/2026_07_27_harries|Harries's second claim page]].