Rational Points of Some Elliptic Curves Related to the Tilings of the Equilateral Triangle

Abstract

Let n be a positive and squarefree integer. We show that the equilateral triangle can be dissected into ncdot k^2 congruent triangles for some k if and only if nle 3, or at least one of the curves C_n :y^2 =x(x-n)(x+3n) and C_{-n} : y^2 =x(x+n)(x-3n) has a rational point with yne 0. We prove that if p is a positive prime such that pequiv 7 (mod 24), then C_p and C_{-p} do not have such points. Consequently, for these primes the equilateral triangle cannot be dissected into pcdot k^2 congruent triangles for any k.