The Diophantine Equation $x^4+y^4=D^2z^4$ in Quadratic Fields

Tuesday, December 1, 2015 - 2:00pm
MSB 312
Melissa Emory (University of Missouri, Columbia)

A.Aigner proved that except in $\mathbb{Q}(\sqrt{-7})$, there are no nontrivial quadratic solutions to $x^4+y^4=z^4.$ The result was later re-proven by D.K. Faddeev and the argument simplified by L.J. Mordell. This talk discusses work to extend this result that shows that nontrivial quadratic solutions exist to $x^4+y^4=D^2z^4$ precisely when either $D=1$ or $D$ is a congruent number.