Given the positive integers m,n, solving the well known symmetric Diophantine equation xm+kyn=zm+kwn, where k is a rational number, is a challenge. By computer calculations, we show that for all integers k from 1 to 500, the Diophantine equation x6+ky3=z6+kw3 has infinitely many nontrivial (y≠w) rational solutions. Clearly, the same result holds for positive integers k whose cube-free part is n...

A “diophantine equation” is a system of polynomial equations where we restrict the solution set to rational or integral values. The study of diophantine equations is the study of the nature of these integral or rational solution sets. “diophantine geometry” is the study of diophantine equations using the language of algebraic geometry. Specifically, one associates to a diophantine equation the ...

We prove that if (x, y, n, q) 6= (18, 7, 3, 3) is a solution of the Diophantine equation (xn−1)/(x−1) = y with q prime, then there exists a prime number p such that p divides x and q divides p − 1. This allows us to solve completely this Diophantine equation for infinitely many values of x. The proofs require several different methods in diophantine approximation together with some heavy comput...

Let a, b, c be fixed positive integers satisfying a2 + ab + b2 = c with gcd(a, b) = 1. We show that the Diophantine equation a2x+axby+b2y = cz has only the positive integer solution (x, y, z) = (1, 1, 1) under some conditions. The proof is based on elementary methods and Cohn’s ones concerning the Diophantine equation x2 + C = yn. Mathematics Subject Classification: 11D61