Equal Sums of Quartics (In Context with the Richmond Equation) (ax4+by4+cz4+dw4=0)

Abstract

Consider the below mentioned Equation: ax4+by4+cz4+dw4=0---[1]. In section (1) we consider solution's with the condition on the coefficient's of equation[1]. Namely the product (abcd)=square. In section [2] we consider the coefficients of Equation [1], with the product of coefficient's (abcd) not equal to a square. Historically Equation [1] has been studied by Ajai Choudhry, A. Bremner, M.Ulas [ref. 5] in 2014. Also Richmond [ref. 1 & 2] has done some ground work in 1944 & 1948. This paper has gone a step further, by finding many parametric solutions & new small numerical solutions by the use of unique Identities. The identities are unique, because they are of mixed powers (combination of quartic & quadratic variables) which are then converted to only degree four identities. As an added bonus in section [B], we came up with a few quartic (4-1-n ) numerical solutions for (n < 50) by elliptical mean's. A table of numerical solutions for the (4-1-n) Equation arrived at by brute force computer search is also given [ref # 7].