Assess whether even-degree parametric families yield new numerical solutions

Determine whether parametric solutions of even degree for the Diophantine equation x^4 + y^4 = z^4 + w^4 yield any numerical solutions that are not generated by known parametric solutions of odd degree.

Background

The paper studies the classical Diophantine equation x^4 + y^4 = z^4 + w^4 and introduces a method, following Richmond’s tangent-plane approach, to generate new parametric solutions from known ones. Historically, published parametric solutions for this equation have been of odd degree, and Guy’s account suggests all nonsingular solutions have odd degree, though singular solutions of even degree may exist.

Using this generation method, the authors construct three explicit parametric families of even degrees 74, 88, and 132. They then pose an explicit unresolved question about the novelty of numerical solutions produced by these even-degree families compared to those obtainable from the previously known odd-degree families.