Symmetric Diophantine Equations

Abstract

In this paper we use certain properties of rational binary forms to solve several diophantine equations of the type f(x, y) = f(u, v). If on applying the nonsingular linear transformation T defined by x = αu + βv, y = γu + δv, the binary form φ(x, y) becomes a scalar multiple of the form φ(u, v), we call φ(x, y) an eigenform of the linear transformation T . If f(x, y) = L(x, y)φ(x, y) where φ(x, y) is an eigenform of the linear transformation T and L(x, y) is not an eigenform of T , the diophantine equation f(x, y) = f(u, v) reduces, on making the substitution x = m(αu+βv), y = m(γu+ δv), to a linear equation in the variables u and v while m is an arbitrary parameter. The solution of this linear equation readily yields a parametric solution of the original diophantine equation. We first use eigenforms to obtain parametric solutions of several general types of diophantine equations such as L1(x, y)Q1(x, y)Q s 2(x, y) = L1(u, v)Q r 1(u, v)Q s 2(u, v) and {Πi=1Li(x, y, z)}Qr(x, y, z) = {Πi=1Li(u, v,w)}Qr(u, v,w) where Ls and Qs denote linear and quadratic forms and r and s are arbitrary integers, and then we obtain parametric solutions of several specific diophantine equations such as the equation f(x, y) = f(u, v) where f(x, y) = xn + xn−1y + · · · + yn, n being an arbitrary odd integer and the equation x7 + y7 + 625z7 = u7 + v7 + 625w7.