Using a theorem on linear forms in logarithms, we show that the equation p x À 2 y ¼ p u À 2 v has no solutions ðp; x; y; u; vÞ with xau; where p is a positive prime and x; y; u; and v are positive integers, except for four specific cases, or unless p is a Wieferich prime greater than 10 15 : More generally, we obtain a similar result for p x À q y ¼ p u À q v 40 where q is a positive prime, qc1 mod 12: We solve a question of Edgar showing there is at most one solution ðx; yÞ to p x À q y ¼ 2 h for positive primes p and q and positive integer h: Finally, we use elementary methods to show that, with a few explicitly listed exceptions, there are at most two solutions ðx; yÞ to jp x 7q y j ¼ c and at most two solutions ðx; y; zÞ to p x 7q y 72 z ¼ 0; for given positive primes p and q and integer c: r
For relatively prime integers a and b both greater than one and odd integer c, there are at most two solutions in positive integers (x, y, z) to the equation a x + b y = c z . There are an infinite number of (a, b, c) giving exactly two solutions.
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