Formalization of Amicable Numbers Theory

2026-01-12

AI Summary

This work presents the first complete formalization of amicable number theory in Lean 4, addressing a longstanding gap in formal proof systems. It introduces rigorous definitions of proper divisors, the sum-of-divisors function, and amicable pairs, and formally verifies Thābit’s formula, Euler’s generalization, and—novelly—the Borho–Hoffmann multiplication method, the latter constituting its first machine-checked proof. The development extends to sociable and betrothed numbers, culminating in a modular 2,076-line codebase comprising 139 theorems. Leveraging tactics such as zify and ring, the framework supports reasoning about divisor sums, multiplicativity, and coprimality, enabling the verification of Poulet’s fifth-order sociable cycle, classical amicable pairs, and a lower bound (>10⁶⁵) for coprime amicable pairs. The library is structured for seamless integration into Mathlib.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Agreement, Argumentation & Negotiation

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Systems and Infrastructure for Web, Mobile and WoT: Decentralized Web and Fediverse systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

This paper presents a formalization of the theory of amicable numbers in the Lean~4 proof assistant. Two positive integers $m$ and $n$ are called an amicable pair if the sum of proper divisors of $m$ equals $n$ and the sum of proper divisors of $n$ equals $m$. Our formalization introduces the proper divisor sum function $\propersum(n) = \sigma(n) - n$, defines the concepts of amicable pairs and amicable numbers, and computationally verifies historically famous amicable pairs. Furthermore, we formalize basic structural theorems, including symmetry, non-triviality, and connections to abundant/deficient numbers. A key contribution is the complete formal proof of the classical Th\={a}bit formula (9th century), using index-shifting and the \texttt{zify} tactic. Additionally, we provide complete formal proofs of both Th\={a}bit's rule and Euler's generalized rule (1747), two fundamental theorems for generating amicable pairs. A major achievement is the first complete formalization of the Borho-Hoffmann breeding method (1986), comprising 540 lines with 33 theorems and leveraging automated algebra tactics (\texttt{zify} and \texttt{ring}) to verify complex polynomial identities. We also formalize extensions including sociable numbers (aliquot cycles), betrothed numbers (quasi-amicable pairs), parity constraint theorems, and computational search bounds for coprime pairs ($>10^{65}$). We verify the smallest sociable cycle of length 5 (Poulet's cycle) and computationally verify specific instances. The formalization comprises 2076 lines of Lean code organized into Mathlib-candidate and paper-specific modules, with 139 theorems and all necessary infrastructure for divisor sum multiplicativity and coprimality reasoning.

Problem

• amicable numbers
• formalization
• Thābit formula
• sociable numbers
• divisor sum

Innovation

• formalization
• amicable numbers
• Lean 4
• Borho-Hoffmann method
• Thābit formula