Problem

Source: Simon Marais Mathematics Competition 2023 Paper B Problem 4

Tags: number theory



(The following problem is open in the sense that the answer to part (b) is not currently known.) Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a,b)$ of positive integers for which there exists a positive real number $r$, such that $$r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}$$are both rational numbers. Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a,b)$ of positive integers for which there exists a real number $r$, such that $$r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}$$are both rational numbers.