A Quartic Diophantine Equation

Abstract

Where a, b, c, d, e 0 are given integers, has engaged the attention of mathematicians for several hundred jeans as can be seen from . The problem was proposed in 1865 as a prize subject by the Accad Nuovi Lincei of Rome . All the methods which have been proposed deduce rational solutions of the above equation from one or more initial solutions.

Finding integer solutions of the above equation is a much more difficult problem and it had seemed impossible to find all of them except in particular eases. From Siegels Theorem it 5 known that there are only a finite number of integer solutions,possibly none, of the above equation when the left hand side has no squared linear factors. Beyond this very little is known about the equation and it is very surprising that a variety of special methods, some very complicated have been devised to deal with the above equation.