2025/04/28 by Marius-Constantin Dinu, Dinu, Marius-Constantin · 1 voice
Computer Science · #FOS: Computer and information sciences #Machine Learning (cs.LG) #Matrix Theory and Algorithms #Symbolic Computation (cs.SC) #cs.LG #cs.SC
paper · pdf · doi:10.48550/arxiv.2505.00730
openalex publication_date 2025/04/28 · arxiv published 2025/04/28 · arxiv updated 2025/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper presents a novel primality test based on the eigenvalue structure of circulant matrices constructed from roots of unity. We prove that an integer n > 2 is prime if and only if the minimal polynomial of the circulant matrix Cn = Wn + Wn2 has exactly two irreducible factors over ℚ. This characterization connects cyclotomic field theory with matrix algebra, providing both theoretical insights and practical applications. We demonstrate that the eigenvalue patterns of these matrices reveal fundamental distinctions between prime and composite numbers, leading to a deterministic primality test. Our approach leverages the relationship between primitive roots of unity, Galois theory, and the factorization of cyclotomic polynomials. We provide comprehensive experimental validation across various ranges of integers, discuss practical implementation considerations, and analyze the computational complexity of our method in comparison with established primality tests. The visual interpretation of our mathematical framework provides intuitive understanding of the algebraic structures that distinguish prime numbers. Our experimental validation demonstrates that our approach offers a deterministic alternative to existing methods, with performance characteristics reflecting its algebraic foundations.