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On the diophantine equation xp - x = yq - y

We consider the diophantine equation xp - x = yq - y, in integers (x, p, y, q). We prove that for given p and q with 2 [less than or equal] p alt; q (*) has only finitely many solutions. Assuming the abc-conjecture we can prove that p and q are bounded. In the special case p = 2 and y a prime power we are able to solve (*) completely.