Abstract
In this paper, we investigate the non-negative integer solutions of the equation of the form qx + py = z2 for x, y and z, with p, q primes. In particular, we consider the equation 3x + py = z2, with p a prime congruent to 5 modulo 12. We prove that (1, 0, 2) is the unique non-negative integer solution of this equation. Moreover, we prove that (1, 0, 2) is also the unique non-negative integer solution for the equation 3x + by = z2 where b is a positive integer congruent to 1 modulo 4 and has a prime factor congruent to 5 modulo 12 or congruent to 7 modulo 12.
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