An Elementary Algorithm for Solving a Diophantine Equation of Degree Fourth with Runge’s Condition

Abstract

We propose an elementary algorithm for solving the diophantine equation (p(x; y) + a1x + b1y)(p(x; y) + a2x + b2y)- dp(x; y)- a3x - b3y -c = 0 ( *) of degree fourth, where p(x; y) denotes an irreducible quadratic form of positive discriminant and (a1; b1) ̸= (a2; b2). The last condition guarantees that the equation ( ) can be solved using the well known Runge’s method, but we prefer to avoid the use of any power series that leads to upper bounds for solutions useless for a computer implementation.