**1. Introduction**

The evolution of Algorithms from a simple route, to complexified paths requires maps from zones of optimal utilization, to be solved sufficiently, in a given amount of time.

These algorithms are constructed for the purpose of building and advancing a continuity for the next location of optimal utilization, in order to realize the importance to form workable nodes and circuits, that are discrete and exact algorithm criteria in a time-basis. Therefore, a complete network on a nonlinear surface and related machine learning epochs is built.

These criteria are based on Fermat's Theorem proving global extrema locations either at stationary or bounding points, based ultimately upon the Pythagorean Theorem, where:

Let **N** be the set of natural numbers 1, 2, 3, … , let **Z** be the set of integers 0, �1, �2, … , and let **Q** be the set of rational numbers *a*/*b*, where *a* and *b* are in **Z** with *<sup>b</sup>* 6¼ 0. In what follows we will call a solution to *xn* <sup>+</sup> *<sup>y</sup> <sup>n</sup>* = *z<sup>n</sup>* where one or more of *x*, *y*, or *z* is zero a *trivial solution*. A solution where all three are non-zero will be called a *non-trivial* solution [1].
