Tiling a triangle into a prime number of congruent triangles

Abstract

We show that, apart from a few known exceptions, if a triangle is cut into $N$ congruent triangles, then $N$ is not a prime number. The exceptions are cutting an isosceles triangle in half, cutting an equilateral triangle in three, a 3-tiling of a 30-60-90 triangle, and the long-known biquadratic tilings of a certain right triangle, when $N$ is a sum of two squares.

Disclosure

“to two similar copies, subdivided quadratically into e2 and f 2 tiles. □ Remark. See Figure 1 for pictures of the exceptional prime tilings. Use of AI We used Claude (Anthropic) for proof-checking and copy-editing. But also Lemma 14 was proved on July 22, 2026 by the LLM, Claude Fable. It was in- vented by Claude in the course of answering the question, whether a triangle with angles (α, 2α, 3β) with α”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T19:06:41.616124Z

Structural counts