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arXiv · 1406.5684

Perfect Numbers and Fibonacci Primes (II)

Tianxin Cai·Liuquan Wang·Yong Zhang
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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.

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Tianxin Cai, Liuquan Wang, Yong Zhang. 2014-06-22. Perfect Numbers and Fibonacci Primes (II). https://arxiv.org/abs/1406.5684

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