Triangular numbers and elliptic curves

Abstract

Some arithmetic of elliptic curves and theory of elliptic surfaces is used to find all rational solutions (r, s, t) in the function field Q(m, n) of the pair of equations r(r+l)/2 = ms(s + l)/2 r(r + l)/2 = nt(t + l)/2. It turns out that infinitely many solutions exist. Several examples will be given. © 1996 Rocky Mountain Mathematics Consortium.