Beeson 2026 solution erdos problem 633

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corollary_2: Shows that non-isosceles triangles admitting a nonsquare tiling have only countably many similarity classes.

lemma_18: Sends rational solutions of an even quartic to rational points of an elliptic curve.

lemma_24: Restricts a rational squared tangent at a rational multiple of pi to four values.

proposition_10: Parametrizes the rational-side condition by a rational tangent of a half-angle.

proposition_13: Identifies the tile angles and rationality conditions for every non-isosceles non-reptiling.

proposition_19: Rules out simultaneous squares a squared plus ab plus b squared and a times a plus b.

proposition_20: Shows that the product of t squared minus two and t squared minus three is never a rational square.

proposition_21: Excludes rational square values of the area factor for the doubled-angle Group 2 family.

proposition_22: Excludes rational square values of the area factor for the final Group 2 family.

proposition_26: Gives a nonsquare tiling when a triangle has a 60-degree angle and the stated rational half-angle parameter.

proposition_27: Shows that every Group 1 tiling of the triangle with angles alpha, twice alpha, twice beta is nonsquare.

proposition_28: Excludes square counts in tilings with large angles alpha, twice alpha, and three times beta.

proposition_29: Determines square versus nonsquare counts in the family C equals A over two plus B.

proposition_30: Gives nonsquare tilings in the family C equals twice A plus B over two.

proposition_31: Excludes square counts for the half-angle tile of a triangle with a 60-degree angle.

proposition_9: Expresses the side ratios of a triangle with 3 alpha plus 2 beta equal to pi.

theorem_1: Classifies exactly the triangles that can be cut into a nonsquare number of congruent triangles.

theorem_11: Reduces a non-reptile tiling of a non-isosceles triangle to six angle patterns.

theorem_3: Restricts a square tiling by a nonsimilar tile to isosceles triangles or the triquadratic square family.

theorem_32: Records the paper's one-or-two-tile classification with its unresolved uniqueness-proof scope.

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Michael Beeson, Miklós Laczkovich, and Yan X. Zhang, Solution of Erdős Problem 633, arXiv:2604.03609v3.

Source version

The canonical PDF is the 33-page version submitted to arXiv on 2026-08-25, with a printed date of 2026-08-26. Its printed and PDF page numbers agree. The arXiv record, lists v1 on April 4, v2 on May 4, and v3 on August 25; v3 remains the latest listed version. The arXiv record (https://arxiv.org/abs/2604.03609, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

The first version had 21 pages and the same eight-family main theorem. The present version adds the explicit square-exception Theorem 3, the second $60$-degree tile Proposition 31, and the secondary uniqueness Theorem 32, with revised labels and more examples. Its Proposition 13 distinguishes the first row's quarter-angle rationality condition from the second row's half-angle condition. Citations here use v3's labels; the April forum announcement and earlier PDF are not substituted for it.

Main results and method

Theorem 1 gives the complete eight-family classification of triangles admitting nonsquare tilings. Its complement answers Problem 633. Corollary 2 shows that, beyond the isosceles triangles, there are only countably many similarity classes in those families. Theorem 3 restricts square non-reptilings to isosceles triangles and one rational triquadratic family.

The proof uses Laczkovich's classification to reduce non-isosceles non-reptilings to six angle patterns, then uses rationality of tile sides and area ratios to constrain the tile count. Four nonsquare arguments use rational points on rank-zero elliptic curves; the second $60$-degree tile is excluded by a congruence modulo $3$. Group 1 counting formulas come from Beeson's earlier tiling equations. The nonsquare triquadratic criterion depends only on the square class of $2K^2-M^2$, not on a claim that each representation $M/K$ supplies that exact tile count.

The linked result pages give complete rewritten deductions for the main classification and Theorem 3, including essential same-paper lemmas. External tiling, rationality, trigonometric, and elliptic-curve inputs are stated and cited explicitly. In particular, the paper's four rank calculations are not printed descents: the exact rank and torsion data are imported from the identified LMFDB records. No local Lean build or independent elliptic-curve rank computation is claimed.

Source qualifications

• Theorem 1's necessity proof on p. 9 misidentifies the reference supplying its triquadratic equation. Proposition 29 identifies the applicable theorem in arXiv:1206.2229v3.
• The added p. 16 minimum-count remark attributes a divisibility necessity to that external theorem which its statement and proof do not supply. This extra claim is left unproved; the square criterion does not use it.
• Theorem 32 records the secondary uniqueness statement with a proof pointer and an unresolved angle-identification step on p. 21. It is not a dependency of either main theorem.

Earlier constructions and further coverage

Section 7, pp. 21–32, gives diagrams for all eight families. These illustrate the existence results rather than replacing their cited proofs. Refining dissections into similar triangles and parallelograms to congruent tiles is Laczkovich's method; Herdt's parallelogram rearrangement can reduce counts substantially. Figures 9 and 10 give different constructions with $7007$ and $3575$ tiles of sides $(3,5,7)$. The latter follows Zhang's construction paper. Figures 10 and 11 use different tile shapes for the same large triangle.

These constructive refinements concern which triples $(T,R,N)$ occur, and connect the source to Problem 634. They are recorded here as illustration and proof pointers; their full construction algorithms and exact counts are not independently reconstructed. The source explicitly states that Laczkovich's earlier existence results suffice for the main proof. A separate full classification of attainable counts is not claimed.

Bears on. Problem 633; Problem 634 for construction methods.