Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Claims
1991_08_01_snover_waiveris_williams: Snover, Waiveris and Williams classify the cuttings of a triangle into N congruent copies similar to itself, so the values n^2 + m^2 and 3n^2 occur; a refereed paper in Discrete Mathematics.
2009_01_01_soifer: Soifer shows that some triangle can be cut into 2n^2, 3n^2, 6n^2 and n^2 + m^2 congruent triangles, the constructions the site's remarks credit and the later manuscripts call classical; a book chapter.
2018_11_23_beeson: Beeson proves that no triangle can be cut into seven congruent triangles, nor into eleven; the result the site's remarks credit through his talk slides; a preprint without a journal record.
2025_12_27_zhang: Zhang proves that for an integer-sided tile with sides a, b and c = sqrt(a^2 + ab + b^2) an equilateral triangle can be cut into m^2 ab congruent copies for every m above an explicit threshold; a preprint.
2026_06_27_bonfioli: A manuscript excluding every prime 7 mod 12, 19 among them, determining every n up to 80 by exact search, and leaving the primes 11 mod 12 open under an unproved hypothesis; disputes several published exclusions.
2026_07_17_george: A write-up posted to the site's proof-claims thread asserting that no triangle can be cut into 19 or 46 congruent triangles, nor into p of them for any prime p > 3 with p congruent to 3 mod 4; unreviewed.
2026_07_24_harries: A manuscript posted to the site's proof-claims thread asserting that no triangle is tiled by p congruent triangles for a prime p > 3 congruent to 3 mod 4, so the prime counts are 2, 3 and the primes 1 mod 4; unreviewed.
2026_07_26_beeson: A preprint asserting that a triangle cut into N > 3 congruent triangles, with N prime, is a reptiling, so that the prime counts are exactly 2, 3 and the primes 1 mod 4; one of three contemporaneous prime proofs; unreviewed.
2026_07_27_harries: A manuscript giving exact certificates that 88 and 189 occur, so that 21n^2 occurs exactly for n at least 2, and a computer-assisted, independently certified exclusion of 33; unreviewed.