On the common factors of \(2^n-3\) and \(3^n-2\) (Q5933546)

The author investigates the problem of finding the GCD, \(g\), of \(2^n-3\) and \(3^n-2\). If one calculates \(g\) for \(n\leq 3000\) one finds that \(g=1\) if \(n\equiv 0,1,2\pmod 4\) and \(g=5\) if \(n\equiv 3\pmod 4\). But this is not always true. In fact the author gives a proof of the fact that \(26665\mid g\) iff \(n\equiv 3783\pmod {5332}\). The question of whether \(g\) always equals 26665 in this case remains open, though it is true for \(n\leq 536983\).