Question:Which natural numbers are of the form $a^2 - b^2 - c^2$ with $a>b+c$?

This question came up in (Eike Hertel, Christian Richter, *Tiling Convex Polygons with Congruent Equilateral Triangles*, Discrete Comput Geom (2014) 51:753–759), where it was shown that numbers of this form are numbers of equilateral triangles (of the same size) that can tile a convex pentagon.

It was shown (and this is elementary), that any number not of this form must be an idoneal number and thus be among

$$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848$$

or equal to one of at most two more unknown numbers (which are at least $\geq 2.5 \cdot 10^{10}$). Interestingly, some of these known idoneal numbers can be written as in the question, for example $7=3^2-1^2-1^2$, but most of them can't.

It is known that if the Generalized Riemann Hypothesis holds, then the list above (without the two unknown numbers) is a complete list of idoneal numbers. Then one can easily check which ones can be of the form as in the question. Hence, assuming the conjecture, the question can be answered. Anyhow, my hope was that maybe the question is more elementary to answer.

Question:Can one answer the above question without using the Generalized Riemann Hypothesis?