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

2026-01-25

AI Summary

This study investigates integer solutions to linear Diophantine equations of the form $a\sigma(n) = bn + c$ within a prescribed upper bound $U$, where $\sigma(n)$ denotes the sum-of-divisors function. We propose an efficient algorithm grounded in recursively enumerable sets, implemented in SageMath with native support for MapReduce-style parallelization—the first application of such a framework to systematic searches of this class of number-theoretic equations. Our approach substantially enhances the efficiency of traversing large solution spaces, yielding new solutions to several previously studied equations. These results fill existing gaps in sequences related to superperfect and $f$-perfect numbers and significantly extend the known upper bounds for the existence of quasiperfect and almost perfect numbers.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Diophantine equation
• sum of divisors
• bounded solutions
• perfect numbers
• computational number theory

Innovation

• Diophantine equations
• sum of divisors
• recursive enumeration
• MapReduce parallelization
• SageMath