SIMON-MARAIS 2025 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) Determine all triples of integers $(x, y, q)$ that satisfy the equation

$$x^3 + x^2 + x + 1 = y^q,$$

where $q \geq 3$ is odd.

(b) Determine all quadruples of integers $(x, y, n, q)$ that satisfy the equation

$$x^n + x^{n-1} + \cdots + x + 1 = y^q,$$

where $n, q \geq 2$.