  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.

Number Theory