On squares in certain Lucas sequences (Q1320519)

Let \(a\) be an integer \(\geq 3\). The Lucas sequence \(u_ n= (\alpha^ n- \beta^ n)/ (\alpha-\beta)\), for \(\alpha= (a+ \sqrt{a^ 2-4})/2\) and \(\beta= (a- \sqrt{a^ 2-4})/2\), is considered. It is proved that for \(a\geq 4\), \(u_ n\) is neither a square, nor a double or a triple square, nor six times a square for \(n>3\), except for \(a=338\) and \(n=4\).