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

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