Computing bounded solutions to linear Diophantine equations with the sum of divisors

TL;DR

This paper introduces an efficient computational method to find all solutions to a specific linear Diophantine equation involving the sum of divisors, leveraging SageMath and parallelization to explore large solution spaces.

Contribution

The authors develop a novel, efficient computational approach for solving a class of linear Diophantine equations, enabling discovery of new solutions and extending bounds in open problems.

Findings

Discovered new solutions to various equations involving sum of divisors.

Closed gaps in known solutions for hyperperfect and f-perfect numbers.

Significantly increased bounds for quasiperfect and almost-perfect numbers.

Abstract

We propose an efficient computational method for finding all solutions $n\leq U$ to the Diophantine equation $a\sigma(n) = bn + c$, where integer coefficient $a,b,c$ and an upper bound $U$ are given. Our method is implemented in SageMath computer algebra system within the framework of recursively enumerated sets and natively benefits from MapReduce parallelization. We used it to discover new solutions to many published equations and close gaps in between the known large solutions, including but not limited to hyperperfect and $f$-perfect numbers, as well as to significantly lift the existence bounds in open questions about quasiperfect and almost-perfect numbers.