A Note on Quartic Equations with only Trivial Solutions

Abstract

In our notice we propose the classification of some quartic equations with only trivial solutions by the auxiliary equations. For proving trivial solutions of the quartic equations we use method infinite descent based on the number of prime integers of the solution of the auxiliary equation. The quartic equations with only trivial solutions belong to "higher arithmetic". The main tools of their investigation are the fundamental theorem of arithmetic, infinite descent and elliptic curves. In 1969, L.J. Mordell has observed that the triviality of the solution of $x^4 + y^4 = 2z^2$ implies the triviality of the solution of $x^4 + 6x^2y^2 + y^4 = z^2$. In his treatise, he did not study the reason or the mechanics of this relationship nor did he conjecture that this link extended to other fourth-degree diophantine equations. We have discovered that the connection identified by Mordell can be described by using the concept "resolvents". In present notice which carries an expository character, we show our findings by presenting the reader with a very easy to follow and transparent example of how the proposed technique works.