Exploring the Diophantine equation Ax'4 + By'4 = Cz'4 in quadratic fields

A. Aigner proved in 1934 that, except in Q( -7 ), there are no nontrivial quadratic solutions to the Diophantine equation x4 + y4 = z4. The result was later re-proven by D.K. Faddeev and the argument was simplified by L.J. Mordell. This paper extends this result to certain other Diophantine equations of the form Ax 4 + By4 = Cz 4. Our main result proves that nontrivial quadratic solutions exist to X4 + Y4 = C2Z4 precisely when either C = 1 or C is a congruent number. Looking for other families of polynomials with nontrivial solutions, we observe the existence of nontrivial quadratic solutions to x 4 + y4 = Cz 4 when C is of the form 8m 2 − 1 which includes primes on a certain diagonal of the odd variant of the Ulam Spiral. We also observe the existence of solutions to x4 + by4 = z4 when b satisfies a certain antipalindromic polynomial.