Nagell–Ljunggren conjecture

The Nagell–Ljunggren equation concerns integer solutions to

$$y^q=\frac{x^n-1}{x-1}.$$

Nagell–Ljunggren conjecture. The only solutions in integers $x,y>1$, $n>2$, and $q\geq 2$ are

$$\frac{3^5-1}{3-1}=11^2,\qquad \frac{7^4-1}{7-1}=20^2,\qquad \frac{18^3-1}{18-1}=7^3.$$

This is a classical Diophantine conjecture about the perfect-power values of a geometric sum. The statement gives the complete list of presently conjectured solutions; its general resolution remains open.