Binomial coefficients and Lucas sequences (Q1604986)

Let sequences \(\{u_n\}_{n\geq 0}\) and \(\{v_n\}_{n\geq 0}\) be defined by \(u_n= \frac{a^n-b^n}{a-b}\), \(v_n= a^n+b^n\) where \(a,b\) are integers such that \(a>|b|\). (Such sequences are Lucas sequences such that the associated quadratic polynomial has integer roots. It is not mentioned that the degenerate case \(b=0\) should be excluded.) The authors investigate the Diophantine equations: \[ u_n= \binom {m}{k}, \qquad v_n= \binom {m}{k} \] where \(m\geq 2k> 2\). Their main result is that if \(k\) is fixed, then the latter equation has at most finitely many solutions. With a few exceptions, the same is true for the former equation. The article concludes with no less than 44 references.