The Diophantine Equation $x^4 + 2 y^4 = z^4 + 4 w^4$---a number of improvements

TL;DR

This paper reports that, using improved algorithms, the only known non-trivial solution to the specific Diophantine equation within a large bound is unique, confirming no others exist up to 100 million.

Contribution

The authors developed enhanced algorithms that verified the uniqueness of the known solution to the Diophantine equation within a large numerical bound.

Findings

Confirmed the known solution is unique up to 100 million

Improved computational methods increased efficiency

Established a new benchmark for solving similar equations

Abstract

The quadruple $(1\,484\,801, 1\,203\,120, 1\,169\,407, 1\,157\,520)$ already known is essentially the only non-trivial solution of the Diophantine equation $x^4 + 2 y^4 = z^4 + 4 w^4$ for $|x|$, $|y|$, $|z|$, and $|w|$ up to one hundred million. We describe the algorithm we used in order to establish this result, thereby explaining a number of improvements to our original approach.