An Analytical Study of Sums of Two Squares and Powers of Positive Integers in Relation to the Hasse Principle

Abstract

In this research paper aims to study the sum of Two squares and powers of positive integers  which cannot be written in the mathematical form  for all values of  for the positive integers u, v, z. It also aims to study a number of example that are consistent with its principle or are considered counterexamples to Hasse Principle, as well as the strong approximation of the results for positive integers resulting from the aforementioned mathematical formula. And also by linking the existence of solutions to the Diophantine equations on rational numbers , And also solutions for complements of rational numbers .