Solutions to Diophantine Equations

Abstract

. In this article, we investigate the solutions of the Diophantine equations p x + ( p + 1) y + ( p + 2) z = M 2 for primes p ≥ 2 when 1 ≤ x , y , z ≤ 2. We establish : (i) When p = 2 and x = y = z = 1, the equation has a unique solution. (ii) When p = 4 N + 1 and 1 ≤ x , y , z ≤ 2, the equations have no solutions. (iii) When p = 4 N + 3 and x = y = z = 1, the equation has infinitely many solutions. (iv) When 3 ≤ p ≤ 199 and x = 1, y = z = 2, the equation has exactly one solution. (v) In all other cases 1 ≤ x , y , z ≤ 2 which are not mentioned above, the equations have no solutions.