Solving diophantine equations x4+ y4= q zp

The Diophantine Equation x4 ± y4 = iz2 in Gaussian Integers

The Diophantine equation x4 ± y4 = z2, where x, y and z are integers was studied by Fermat, who proved that there exist no nontrivial solutions. Fermat proved this using the infinite descent method, proving that if a solution can be found, then there exists a smaller solution (see for example [1], Proposition 6.5.3). This was the first particular case proven of Fermat’s Last Theorem (which was ...

The Diophantine Equation x4 + 2 y4 = z4 + 4 w4

Solving Linear Diophantine Equations

An overview of a family of methods for nding the minimal solutions to a single linear Diophantine equation over the natural numbers is given. Most of the formal details were dropped, some illustrations that might give some intuition on the methods being presented instead.

Explicit Methods for Solving Diophantine Equations

We give a survey of some classical and modern methods for solving Diophantine equations.

Fast Methods for Solving Linear Diophantine Equations

We present some recent results from our research on methods for finding the minimal solutions to linear Diophantine equations over the naturals. We give an overview of a family of methods we developed and describe two of them, called Slopes algorithm and Rectangles algorithm. From empirical evidence obtained by directly comparing our methods with others, and which is partly presented here, we a...