Perfect numbers and Fibonacci primes

Abstract

In this paper, we introduce the concept of $F$-perfect number, which is a positive integer $n$ such that ${\sum }_{d|n,d<n}{d}^2=3n$. We prove that all the $F$-perfect numbers are of the form $n=F_{2k-1}{F}_{2k+1}$, where both $F_{2k-1}$ and $F_{2k+1}$ are Fibonacci primes. Moreover, we obtain other interesting results and raise a new conjecture on perfect numbers.

9 pages, to appear in International Journal of Number Theory