Perfect Numbers and Fibonacci Primes (II)

TL;DR

This paper explores a specific Diophantine equation related to perfect numbers and Fibonacci primes, establishing a link to the twin primes conjecture and characterizing solutions involving Fibonacci primes.

Contribution

It characterizes solutions to a Diophantine equation involving Fibonacci primes and connects the twin primes conjecture to the solvability of a related equation.

Findings

Solutions are Fibonacci prime products for certain parameters.

The twin primes conjecture is equivalent to the infinite solutions of a specific equation.

Finitely many solutions exist outside the characterized Fibonacci prime products.

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.