Non-Homogeneous Solution of Quintic Equation with Five Unknowns P4-q4 = 82(l2-m2)σ3

Abstract

The non-zero unique integral solutions of the Quintic Diophantine equation with five unknowns, expressed as p⁴-q⁴=82(l²-m²)σ³ can be obtained by applying the linear transformations p=u+v, q=u-v; and m=2u-v (u≠v≠0). Likewise, additional changes can yield several non-zero integer solutions, resulting in multiple integral solutions to the problem. Numerous remarkable relationships between the responses and a few distinct numbers are displayed.