On the diophantine equation \(x^p-x=y^q-y\)
The authors consider the diophantine equation \(x^p-x= y^q-y\) in integers \((x,p,y,q)\). They show that this equation has only finitely many solutions when some values among \(x\), \(y\), \(p\) and \(q\) are fixed. For example using the result of Davenport, Lewis, and Schinzel they show that for a given \(p\) and \(q\) with \(2\leq p< q\), the above equation has only finitely many integral solutions. Assuming the \(abc\)-conjecture they show that \(p\) and \(q\) are bounded. Thus under the \(abc\)-conjecture, the above equation has only finitely many solutions. Further they show that the curve \(C\) defined by the equation has only finitely many rational points \((x,y)\) when \(2\leq p< q\) and \(q\geq 4\). In the special case \(p=2\) and \(y\) a prime power they solved the equation completely.
scientific article; zbMATH DE number 1331812