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 iis a positive integer);where the variables on the LHS are less than or equal to the corresponding variable on the RHS. The proofs are within the context of Sub-Rings.The common factor is that each of the variables x,y,z,v,u, dXYZV and dXYZVUare multiples of (n-f), where nand fare real numbers. The solutions derived herein can be extended to other problems wherein (n-f) can take the form of polynomials such as (3a-1), (12b-5), (as-bs), etc..

Keywords: Markoff Equation; Prime Numbers; Dynamical Systems; Sub-Rings and Ring-Theory; Mathematical Cryptography; Beal Conjecture; Analysis; Diophantine Equations; Nonlinearity