Perfect numbers and Fibonacci primes (III)

Abstract

In this article, we consider the Diophantine equation ${\sigma }_2(n)-n^2=An+B$ with $A=P^2\pm 2$. For some $B$, we show that except for finitely many computable solutions in the range $n\le (|A|+|B|{)}^3$, all the solutions are expressible in terms of Lucas sequences. Meanwhile, we obtain some results relating to other linear recurrent sequences.

11 pages