Primality Testing via Circulant Matrix Eigenvalue Structure: A Novel Approach Using Cyclotomic Field Theory

TL;DR

This paper introduces a new deterministic primality test based on the eigenvalue structure of circulant matrices derived from roots of unity, connecting cyclotomic field theory with matrix algebra for practical primality verification.

Contribution

It establishes a novel primality criterion using minimal polynomial factorization of circulant matrices, integrating cyclotomic fields and eigenvalue analysis for the first time.

Findings

Eigenvalue patterns distinguish primes from composites

The method provides a deterministic primality test

Experimental validation shows competitive performance

Abstract

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 $C_n = W_n + W_n^2$ has exactly two irreducible factors over $\mathbb{Q}$. 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.