Fibonacci and Lucas numbers as difference of two repdigits (Q2154671)

The Fibonacci sequence \((F_k)_{k\geq 0}\) and Lucas sequence \((L_k)_{k\geq 0}\) are given by \(F_0=0,F_1=1\), \(L_0=2,L_1=1\) and \(F_k=F_{k-1}+F_{k-2}\), \(L_k=L_{k-1}+L_{k-2}\) for \(k\geq 2\). The authors consider the problem of finding those Fibonacci numbers and Lucas numbers that can be written as a difference of two repdigits, these are positive integers whose digits in their decimal expansions are all the same. In other words, they consider the Diophantine equations \[ F_k=\frac{d_1(10^n-1)}{9}-\frac{d_2(10^m-1)}{9},\ \ L_k=\frac{d_1(10^n-1)}{9}-\frac{d_2(10^m-1)}{9} \] to be solved in positive integers \(k,m,n,d_1,d_2\) where \(n\geq 2\) and \(d_1,d_2\in\{ 1,\ldots ,9\}\). The authors give the complete lists of solutions to these equations. Their results imply that for Fibonacci numbers one has \(k\leq 11\) and for Lucas numbers \(k\leq 18\). Another consequence is that no Fibonacci number or Lucas number can be expressed as a difference of two different powers of \(10\). The authors obtain their results by carefully applying Matveev's lower bound for linear forms in logarithms of algebraic numbers, together with techniques to deal with the `small' solutions. To dispose of some special cases, the authors use earlier work of themselves and others listing the Fibonacci and Lucas numbers that are the concatenation of two or three repdigits.