On the quinary homogeneous bi-quadratic equation x⁴ + y⁴ - (x + y)w³ = 14<i>_z</i>2<i>T</i> 2

Abstract

The purpose of this paper is to examine the non-zero distinct integral solutions of quinary biquadratic homogeneous diophantine equation x⁴ + y⁴ - (x + y)w³ = 14z²T² in integers. In this paper, we present some different patterns of integral solutions to the above diophantine equation in five variables. Also, we obtain some properties as relations between solutions and special numbers.