Perfect Numbers and Fibonacci Primes (II)

Abstract

In this paper, we study the diophantine equation ${\sigma }_2(n)-n^2=An+B$. We prove that except for finitely many computable solutions, all the solutions to this equation with $(A,B)=(L_{2m},F_{2m}^2-1)$ are $n=F_{2k+1}{F}_{2k+2m+1}$, where both $F_{2k+1}$ and $F_{2k+2m+1}$ are Fibonacci primes. Meanwhile, we show that the twin primes conjecture holds if and only if the equation ${\sigma }_2(n)-n^2=2n+5$ has infinitely many solutions.

7 pages, 2 tables. This is an original research article related to Diophantine equations, Fibonacci primes, twin primes and perfect numbers