On the Diophantine equation \(\frac{x^3-1}{x-1}=\frac{y^n-1}{y-1}\) (Q1775568)

The author proves: Theorem. The positive integer solutions \((x,y,n)\) of the equation \[ {x^3-1\over x-1}={y^n-1\over y-1}, \quad \text{with \(n\) odd and \(x\not=y\),} \] are \((x,y,n)=(5,2,5)\) and \((90,2,13)\). The proof uses a previous result of Le and elementary arguments of algebraic number theory for quadratic fields.