On the Equations px − by = c and ax + by = cz

Abstract

This paper is a response to a problem in [R. K. Guy, "Unsolved Problems in Number Theory," Springer-Verlag, New York, 1981] (see Introduction). The main results are the following: The equation px − by = c, where p is prime, and b > 1 and c are positive integers, has at most one solution (x, y) when y is odd, except for five specific cases, and at most one solution when y is even. The equation pn − qm = pN − qM, where p and q are primes, has no solutions (n, m, N, M) unless (p/q) = (q/p) = 1, except for four specific cases. The equation |px − qy| = c has at most two solutions except for three specific cases. The equation ax + by = pz, where a > 1, b > 1, (a, b) = 1, and p is prime, has at most two solutions when p is odd and at most one solution when p = 2 except for two specific cases.