On Fibonacci and Pell numbers of the form \(kx^2\). (Almost every term has a \(4r+1\) prime factor). (Q2785026)

It was shown that neither the Fibonacci nor the Pell number sequence has terms of the form \(px^2\) for prime \(p\equiv 3\pmod 4\), with one exception in each sequence. The main idea of showing this can be used to prove a stronger result; namely, that, with a small number of exceptions, neither sequence has terms of the form \(kx^2\) if \(k\) is an integer all of whose prime factors are congruent to 3 modulo 4. An interesting corollary is that, with 11 exceptions, every term of the Fibonacci sequence has a prime factor of the form \(4r+1\) and, similarly, with 5 exceptions, for the Pell sequence.