An Algorithm for Solving a Quartic Diophantine Equation Satisfying Runge’s Condition

Abstract

In this paper we suggest an implementation of elementary version of Runge's method for solving a family of diophantine equations of degree four. Moreover, the corresponding solving algorithm (in its optimized version) is implemented in the computer algebra system PARI/GP.

Practical solution of some families of quartic and sextic diophantine hyperelliptic equations

“…Thus, if the constant A(F ) is small, we can have a practical algorithm for finding the set C(Z). In [3,8,9] the previous method allows to practically solve some Diophantine equations e.g. x nr + y n = q in [3] or x 4 − x 2 y − xy 2 − y 2 + 1 = 0 in [8].…”

“…In [3,8,9] the previous method allows to practically solve some Diophantine equations e.g. x nr + y n = q in [3] or x 4 − x 2 y − xy 2 − y 2 + 1 = 0 in [8]. So this method, except the uniform bounds that provides, sometimes it may also be appropriate in order to get a practical algorithm for the integer points.…”