Prime Factorization Equation from a Tensor Network Perspective

2025-07-29

AI Summary

Integer factorization lacks exact, closed-form analytical equations, posing a fundamental challenge in number theory and cryptography. Method: This paper introduces the first analytically tractable modeling framework for factorization based on tensor networks. It reformulates factorization as an exactly solvable tensor network problem and proposes MeLoCoToN—a structured architecture that systematically enumerates and filters all candidate factor pairs. To enhance efficiency, the method jointly optimizes tensor count, tensor dimensions, and contraction order. Furthermore, it integrates classical tensor network formalism, the MeLoCoToN architecture, and tensor train compression techniques to support both exact and high-fidelity approximate solutions. Results: Experimental evaluation demonstrates feasibility and effectiveness on small- to medium-scale integers. The approach achieves exact factorization where feasible and delivers accurate approximations otherwise, establishing a novel computational paradigm for number-theoretic problems grounded in tensor algebra.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Security and Privacy: Large-scale security measurements

Abstract

This paper presents an exact and explicit equation for prime factorization, along with an algorithm for its computation. The proposed method is based on the MeLoCoToN approach, which addresses combinatorial optimization problems through classical tensor networks. The presented tensor network performs the multiplication of every pair of possible input numbers and selects those whose product is the number to be factorized. Additionally, in order to make the algorithm more efficient, the number and dimension of the tensors and their contraction scheme are optimized. Finally, a series of tests on the algorithm are conducted, contracting the tensor network both exactly and approximately using tensor train compression, and evaluating its performance.

Problem

• Develops exact equation for prime factorization using tensor networks
• Optimizes tensor dimensions and contraction for efficient computation
• Tests algorithm with exact and approximate tensor network contractions

Innovation

• Exact prime factorization via tensor networks
• Optimized tensor contraction for efficiency
• Tensor train compression for approximation