The Solution for the Non linear Diophantine Equation (7^k -1)^x +(7^k)^y = z² with k as the positive even whole number

Abstract

This research may provide the solutions (if any) from the Non-linear Diophantine equation (7 k — 1) x + (7 k ) y = Z 2.There are 3 possibilities to determine the solutions from the Non-linear Diophantine equation: single solution, multiple solutions, and no solution. The research method is conducted in two stages: first, using a simulation to determine the solutions (if any) from the Non-linear Diophantine equation (7 k — 1) x + (7 k ) y = z 2; second, using Catalan conjecture and characteristics of congruency theory, which is proven that the Non-k linear Diophantine equation has a single solution ( x , y , z ) = ( 1 , 0 , 7 k 2 ) for x, y, z as non-negative whole numbers and k as the positive even whole number.

The Primitive-Solutions of Diophantine Equation x^2+pqy^2=z^2, for primes p,q

“…There are many forms of Diophantine equations with various variables defined. Rahmawati et al [7] figured out the solutions from the equation ( ) ( ) where , , and are non-negative integers and is the positive even integer, Burshtein [4] , where all the variables are integers. Some cases of this problem have been solved, such as for case of (see in [8]).…”