Simultaneous Diophantine Equations

TL;DR

This paper provides parametric solutions to specific quartic Diophantine equations, supporting the conjecture that non-trivial solutions exist where the products of variables are equal on both sides.

Contribution

It introduces explicit parametric solutions for certain quartic equations, advancing understanding of their solution structure and supporting existing conjectures.

Findings

Explicit parametric solutions for (4-3-3), (4-4-4), (4-5-5), (4-6-6) equations

Evidence supporting the conjecture of non-trivial solutions with equal products

Progress in solving specific classes of quartic Diophantine equations

Abstract

This paper gives parametric solutions to quartic equations of the type,(4-3-3),(4-4-4),(4-5-5) and (4-6-6), According to Lander, Parkin, and Selfridge (2) conjecture, there are non-trivial solutions of the quartic equations,(4-3-3),(4-4-4),(4-5-5),(4-6-6), where products of the variables on both sides of the equation are equal.