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].
Highlights
Existence of solution for Diophantine Equation ax4 + by4 + cz4 + dw4 = 0 are known if is square number
Z2 = t2 + (d + 2)t + d + 1, So, we look for the integer solutions {
We show Diophantine Equation, (d2 − 2dm2 + 2m4)X4 + (d2 − 2dm2 + 2m4)Y4 = 2(−d + m2)2Z4 + 2W4 has infinitely many integer solutions. d, m are arbitrary
Summary
We show some parametric solutions of ax4 + by4 + cz4 + dw4 = 0 with abcd is square number. By repeating this process, we can obtain infinitely many integer solutions for Equation (1). By parameterizing the first Equation and substituting the result to second Equation, we obtain quartic Equation below to make (w= square).
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