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Let a, b, c be fixed positive integers satisfying a 2 + ab + b 2 = c with gcd(a, b) = 1. We show that the Diophantine equation a 2x +a x b y +b 2y = c z 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 x 2 + C = y n .doi:10.1017/s0004972713000956 fatcat:65iivzgsjvddvewaihv6nyc3n4