On Some Properties of X2+y2+z2+v2=dxyzv; and X2+y2+z2+v2+u2=dxyzvu, and Xi+yi+zi + vi =dxyzv.

Abstract

This article develops “existence” properties for the equations x2+y2+z2+v2=dXYZV; And x2+y2+z2+v2+u2=dXYZVU, and xi+yi+zi + vi =dXYZV (where i is a positive integer); and the results are applicable where all variables are Integers (ie. proofs within the context of Sub-Rings). Collectively and individually, these equations have wide applications in Computer Science, Physics, Applied Math and Finance/Economics.