Symmetric Diophantine equations (Q1767344)

The author describes a method to obtain infinite parametric solutions of some Diophantine equations of type \(f(x,y)=f(u,v)\) where \(f\) is a form (usually the product of some linear and quadratic forms) with rational coefficients. The main idea is to apply a non-singular linear transformation \(x=\alpha u+\beta v\), \(y=\gamma u+\delta v\) such that the binary form \(\phi (x,y)\) becomes a scalar multiple of \(\phi (u,v)\), that is \(\phi\) is an eigenform of the above linear transformation. Using this method the author gives parametric solutions of several specific Diophantine equations, such as e.g. \(x^7+y^7+625z^7=u^7+v^7+625w^7\).