Some properties of following two groups of equations are developed in this article: 1) x^2+y^2+z^2+v^2=dXYZ, x^i+y^i+z^i +v^i= dXYZ (where i is an integer), x^2+y^2+z^2= gXYZ and x^2+y^2+z^2+v^2+u^2=dXYZ, x^3+y^3+z^3=dXYZ, and x^6+y^6+z^6=dXYZ, x^3+y^3+z^3+x^6+y^6+z^6=dXYZ, [(x^12+y^12+z^12)-(x^6+y^6+z^6)]=dXYZ and x^i+y^i+z^i=dXYZ (i is a positive integer), where x│X (ie. X is a multiple of x), y│Y, and z│Z are real numbers, and each of the variables x,y,z, v, u and dXYZ are multiples of (n-f), and n and f are real numbers. 2) X^2+Y^2+Z^2+V^2= gXYZ, X^2+Y^2+Z^2=dXYZ and X^2+Y^2+Z^2+V^2 +U^2 = gXYZ (all of which are special cases of the above-mentioned equations), where each of the variables x,y,z, v, u and dXYZ are multiples of (n-f), and n and f are real numbers. This article: i) Summarizes the relationships of these equations to Homotopy Theory, PDEs, Mathematical Cryptography and Analysis. ii) Introduces simple Java codes for finding solutions to the equations [(x^12+y^12+z^12)-(x^6+y^6+z^6)]=rXYZ, x^3+y^3+z^3=k, and x^2+y^2+z^2+v^2 +u^2 = rXYZ for integers between zero and 10^1677721600000000 (and even greater positive-integers depending on available computing power). iii) Explains why the “Variance-Inflation-Factor” is invalid. iv) Explains how (n-f) can be used as a measure of Multicollinearity. v) Introduces several Anomaly-Detection indicators (including (n-f)). The proofs developed herein are within the context of Sub-Rings. The solutions derived herein can be extended to other problems wherein (n-f) can take the form of polynomials/functions such as (6d-3), (14c-5), (ai-bi), etc.. Common elements are highlighted, and some of the results are applicable where all variables are Integers
