SIMON-MARAIS 2023 B4

The following problem is open in the sense that the answer to part (b) is not currently known. A solution to part (a) will be awarded 7 points. Up to 7 additional points may be awarded for progress on part (b).

(a) 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

$$r^a + \sqrt{n} \quad \text{and} \quad r^b + \sqrt{n}$$

are both rational numbers.

(b) 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

$$r^a + \sqrt{n} \quad \text{and} \quad r^b + \sqrt{n}$$

are both rational numbers.