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

Abstract

The quadruple $(1484801,1203120,1169407,1157520)$ already known is essentially the only non-trivial solution of the Diophantine equation $x^4+2y^4=z^4+4w^4$ 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.