Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Max A. Alekseyev

TL;DR

This work addresses finding all integers $n\le U$ solving $a\sigma(n)=bn+c$ by modeling the search space as a tree $T_U$ rooted at $1$ and performing a restricted depth-first traversal. The authors introduce several pruning and shortcut techniques, notably a prime-wheel-based pruning strategy and parity/gcd-based case handling, all embedded in a SageMath implementation using the RES framework and MapReduce, enabling scalable, parallel computation. They demonstrate substantial practical gains by discovering new solutions and lifting existence bounds for challenging classes such as quasiperfect, almost-perfect, hyperperfect, and $f$-perfect numbers, with results reported up to very large bounds and made available publicly. The approach also includes configurable constraints to tailor searches and highlights potential extensions to other multiplicative functions, suggesting broad applicability for arithmetic-equation searches beyond the present study.

Abstract

We propose an efficient computational method for finding all solutions $n\leq U$ to the Diophantine equation $aσ(n) = bn + c$, 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.

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

TL;DR

This work addresses finding all integers solving by modeling the search space as a tree rooted at and performing a restricted depth-first traversal. The authors introduce several pruning and shortcut techniques, notably a prime-wheel-based pruning strategy and parity/gcd-based case handling, all embedded in a SageMath implementation using the RES framework and MapReduce, enabling scalable, parallel computation. They demonstrate substantial practical gains by discovering new solutions and lifting existence bounds for challenging classes such as quasiperfect, almost-perfect, hyperperfect, and -perfect numbers, with results reported up to very large bounds and made available publicly. The approach also includes configurable constraints to tailor searches and highlights potential extensions to other multiplicative functions, suggesting broad applicability for arithmetic-equation searches beyond the present study.

Abstract

We propose an efficient computational method for finding all solutions to the Diophantine equation , where integer coefficient and an upper bound 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 -perfect numbers, as well as to significantly lift the existence bounds in open questions about quasiperfect and almost-perfect numbers.
Paper Structure (19 sections, 3 theorems, 11 equations, 1 figure, 2 tables)

This paper contains 19 sections, 3 theorems, 11 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Method outline
  4. Shortcuts
  5. Pruning with prime wheel
  6. Case of odd $\sigma$
  7. Case of $\gcd(a',c')>1$
  8. SageMath implementation
  9. RES framework
  10. Configurations reduction
  11. MapReduce parallelization
  12. Additional constraints
  13. Solutions above $U$
  14. Availability
  15. Applications
  16. ...and 4 more sections

Key Result

Theorem 3.1

For integers $d$ and $\ell>0$, the number $n=2^{\ell-1}(2^\ell-d-1)$ is a solution to $\sigma(n)=2n+d$ whenever $2^\ell - d - 1$ is prime.

Figures (1)

  • Figure 1: The tree $T_U$ for $U=60$, where some nodes $1\cdot p$, $2\cdot p$, and $3\cdot p$ with prime $p$ are hidden under ellipses.

Theorems & Definitions (5)

  • Theorem 3.1: OEIS OEIS
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof