On products of prime powers in linear recurrence sequences (Q6619741)

In this interesting and well written paper, the authors study the diophantine equation \(U_n=p^x q^y\), where \(U=U_n\) is a linear recurrence sequence, \(p, q\) are distinct primes, and \(x, y\) are non-negative integers. They show that under certain assumptions on \(U\), this diophantine equation has at most two solutions, if the primes are outside a finite computable set. It should be noted that when \(U\) is a Lucas Lehmer sequence this result can be obtained readily from the well known primitive divisors theorem. However, the authors apply their result to sequences such as the one below that do not fall in this category. They show that the equation \(F_n=p^x q^y+2\), where \(F_n\) is the Fibonacci sequence, has at most two solutions when \(n\) is greater than \(4\), unless \((p,q)=(2,3)\) or \((2,19)\), in which case the solutions are listed explicitly. The methods used are standard, where Baker's method first yields an upper bound for \(n, x, y\) in terms of \(p,q\). In the next step this typically huge upper bound is reduced using the Baker-Davenport method combined with other methods.