Pentagonal numbers in the associated Pell sequence and Diophantine equations \(x^2(3x-1)^2=8y^2\pm 4\) (Q2746555)

Here the authors prove that the only generalized pentagonal numbers (integers of the form \(m(3m-1)/2\)) in the associated Pell sequence \(Q_{n+2}=2Q_{n+1}+Q_n\), where \(Q_0=Q_1=1\), are 1 and 7. Since \(x=Q_n\) gives all integral solutions of the Diophantine equations \(x^2-2y^2=\pm 1\), all integral solutions of \(x^2(3x-1)^2=8y^2\pm 4\) are obtained as a corollary.