Practical solution of the Diophantine equation \(X^{nr}+Y^n = q\) (Q633470)

The author studies the Diophantine equation \[ X^{nr} + Y^n = q \] for positive integers \(n,r,q\) with \(n \geq 3\) odd. He proves the estimate \(|x| \leq |q|^{1/r}\) for the integer solutions \((x,y)\) of the above equation. This estimate is an immediate consequence of Theorem 1.1 of the paper which states that if \((x,y)\) is an integer solution to the equation \(X^n + Y^n = q\), then \(|x| \leq |q|\). As an application, it is proven that, for \(r\) fixed, the exponential Diophantine equation \(X^{yr} - Y^y = 1\) admits no nontrivial integer solution \((X,Y,y)\) with \(y \geq 2\) and \(X,Y > 0\). The proofs are based on the binomial theorem and some elementary properties of the gamma function. The author also provides an algorithm for the solution of the equation \(X^{nr} + Y^n = q\), and illustrates it by some examples.