Biquadratic Equation with Four Unknowns

Abstract

Number theory relies heavily on diophantine equations, which come in a wide variety of forms.  There are several Diophantine equations with no solution, trivial solutions, a finite number of solutions, and an infinite number of solutions. Among higher degree Diophantine Equations,there are mainly two types of equations.When the degree is four, they are  homogeneous and non-homogeneous bi-quadratic equations. In its most generic form, its integral solution may be required. Since ancient times, both homogeneous and non-homogeneous Biquadratic equations have piqued the curiosity of many mathematicians.  This paper concerns with the problem of determining non-trivial integral solutions of the non-homogeneous Biquadratic equation with four unknowns given by  7xy + 3z²  = 3 w⁴ . Infinitely many non-zero integer solutions of the equation is found by introducing the linear transformations x = u + v, y = u - v, z = v .