Tiling a triangle into a prime number of congruent triangles
Abstract
We prove that if a triangle is cut into N congruent triangles and N is prime, then N=2, N=3, or N≡ 1 pmod 4, and every such N occurs. In particular, if N > 3 then N is not a prime congruent to $3$ modulo $4$. The argument divides into cases according to possible shapes of the tiled triangle. Let α,β,γ be the angles of the tile. Prior to this posting, there remained several open shapes. The two most difficult shapes were (2α,β,α+β), the {\em triquadratic} shape, and (β,β,3α), the {\em isosceles-β} shape, in which 3α+2β= π and α is not a rational multiple of π. A previous version of this paper relied on incorrect proofs of those cases, which have been retracted, and are replaced here by correct proofs. For each of those two shapes the number of tiles has the form N = j2 N0, where N0 > 1 is determined by the shape of the tile and j is a positive integer; so N is prime, or even squarefree, only if j = 1. We prove by geometric reasoning that neither shape admits a tiling with j = 1, which shows that for both shapes N is never prime, and indeed never squarefree. The shape (β,β,3α+β) (where 3α+ 3β= π) has been settled by Bonfioli. In this paper, we settle the remaining cases, by number-theoretic arguments based on the area equation. Those proofs rule out N prime, but not N squarefree, and we give examples to show that tilings with squarefree N do sometimes exist.
Related
- No Prime Tiling of an Isosceles Triangle
- Pascal tiling and congruences modulo N in Pascal's triangle
- Ordered matchings versus triangles via pseudorandom triangle-free graphs
- Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields
- Tunnell-type criteria for variants of the congruent number problem
BibTeX
@misc{beeson2026tiling,
title = {{Tiling a triangle into a prime number of congruent triangles}},
author = {Michael Beeson and Grigore Roşu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.23453},
eprint = {2607.23453},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.23453},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tiling a triangle into a prime number of congruent triangles
AU - Michael Beeson
AU - Grigore Roşu
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2607.23453
UR - https://arxiv.org/abs/2607.23453
AB - We prove that if a triangle is cut into N congruent triangles and N is prime, then N=2, N=3, or N≡ 1 pmod 4, and every such N occurs. In particular, if N > 3 then N is not a prime congruent to $3$ modulo $4$. The argument divides into cases according to possible shapes of the tiled triangle. Let α,β,γ be the angles of the tile. Prior to this posting, there remained several open shapes. The two most difficult shapes were (2α,β,α+β), the {\em triquadratic} shape, and (β,β,3α), the {\em isosceles-β} shape, in which 3α+2β= π and α is not a rational multiple of π. A previous version of this paper relied on incorrect proofs of those cases, which have been retracted, and are replaced here by correct proofs. For each of those two shapes the number of tiles has the form N = j^2 N_0, where N_0 > 1 is determined by the shape of the tile and j is a positive integer; so N is prime, or even squarefree, only if j = 1. We prove by geometric reasoning that neither shape admits a tiling with j = 1, which shows that for both shapes N is never prime, and indeed never squarefree. The shape (β,β,3α+β) (where 3α+ 3β= π) has been settled by Bonfioli. In this paper, we settle the remaining cases, by number-theoretic arguments based on the area equation. Those proofs rule out N prime, but not N squarefree, and we give examples to show that tilings with squarefree N do sometimes exist.
ER -