MathDB
Power of r plus square root n is rational

Source: Simon Marais Mathematics Competition 2023 Paper B Problem 4

October 16, 2023
number theory

Problem Statement

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