Perfect numbers and Fibonacci primes (III)

TL;DR

This paper investigates solutions to a specific Diophantine equation involving the sum-of-divisors function, showing most solutions relate to Lucas sequences and providing bounds and relations to linear recurrent sequences.

Contribution

It characterizes solutions to a particular Diophantine equation in terms of Lucas sequences, with bounds and connections to other linear recurrent sequences.

Findings

Most solutions are expressible via Lucas sequences.

Solutions are finite and computable within a specific range.

Established links between the equation and linear recurrent sequences.

Abstract

In this article, we consider the Diophantine equation $\sigma_{2}(n)-n^2=An+B$ with $A=P^2\pm2$. For some $B$, we show that except for finitely many computable solutions in the range $n\leq(|A|+|B|)^{3}$, all the solutions are expressible in terms of Lucas sequences. Meanwhile, we obtain some results relating to other linear recurrent sequences.