Tiling Convex Polygons with Congruent Equilateral Triangles

We study the sets $mathcal{T}_{v}={m in{1,2,ldots}: mbox{there is a convex polygon in }mathbb{R}^{2}mbox{ that has }vmbox{ vertices and can be tiled with $m$ congruent equilateral triangles}}$ , v=3,4,5,6. $mathcal{T}_{3}$ , $mathcal{T}_{4}$ , and $mathcal{T}_{6}$ can be quoted completely. The complement ${1,2,ldots} setminusmathcal{T}_{5}$ of $mathcal{T}_{5}$ turns out to be a subset of Euler’s numeri idonei. As a consequence, ${1,2,ldots} setminusmathcal{T}_{5}$ can be characterized with up to two exceptions, and a complete characterization is given under the assumption of the Generalized Riemann Hypothesis.