The Diophantine Equation's Primitive Solutions x2+py2=z2 for Prime P in Cryptography

Abstract

In this paper, we determine the primitive-solutions of diophantine equations x²+py²=z² , for positive integers x,y,z and prime p to crypography. Our work is based on the development of the previous results, namely using the solutions of the Diophantine equation x²+y²=z² , and looking characteristics of the solutions of the Diophantine equation x²+3y²=z² and x²+9y²=z² and their algorithms  .