On the Diophantine equation \(p^{x_1} - p^{x_2} = q^{y_1} - q^{y_2}\) (Q1433045)

The main result of this paper is: Theorem. Let \(p\) and \(q\) be distinct prime numbers, \(q\) fixed. Then the equation \[ p^{x_1}- p^{x_2}=q^{y_1}-q^{y_2}, \quad \text{ with}\;x_1\not=x_2, \] has only finitely many positive integer solutions \((p,x_1,x_2,y_1,y_2)\). The author shows also that the ABC conjecture implies that the previous result remains true without the restriction `\(q\) fixed'. The proof is quite involved. The main tool is the theory of lower bounds of linear forms of logarithms of algebraic numbers.