Perfect Numbers and Fibonacci Primes (II)

2014/06/22 by Cai, Tianxin, Wang, Liuquan, Zhang, Yong
#11D25 #FOS: Mathematics #Number Theory (math.NT) #Primary 11B83 #Secondary 11D09

paper · doi:10.48550/arxiv.1406.5684

Abstract

In this paper, we study the diophantine equation σ₂(n)-n²=An+B. We prove that except for finitely many computable solutions, all the solutions to this equation with (A,B)=(L_2m,F_2m²-1) are n=F_2k+1F_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 σ₂(n)-n²=2n+5 has infinitely many solutions.

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