Quartic Equations with Trivial Solutions over Gaussian Integers

Felix Sidokhine

paper

Quartic Equations with Trivial Solutions over Gaussian Integers

arXiv:1607.08648

Abstract

In our work we study the equations of the form $a X^{4} + b X^{2} Y^{2} + c Y^{4} = d Z^{2}$ over Gaussian integers by a method of the resolvents. We study as a new equations $X^{4} + 6 X^{2} Y^{2} + Y^{4} = Z^{2}$ (Mordell's equation over $Z [i]$ ), $X^{4} + 6 (1 + i) X^{2} Y^{2} + 2 i Y^{4} = Z^{2}$ and $X^{4} \pm Y^{4} = (1 + i) Z^{2}$ and give the new proofs of the known theorems on $X^{4} + Y^{4} = Z^{2}$ (Fermat - Hilbert), $X^{4} \pm Y^{4} = i Z^{2}$ (Szabó - Najman).