Primality Testing via Circulant Matrix Eigenvalue Structure: A Novel Approach Using Cyclotomic Field Theory
This provides a novel theoretical framework for primality testing, potentially impacting cryptography and number theory, though it appears incremental as it builds on existing algebraic structures.
This paper tackles the problem of primality testing by proving that an integer is prime if and only if the minimal polynomial of a specific circulant matrix has exactly two irreducible factors over the rationals, connecting cyclotomic field theory with matrix algebra to develop a deterministic primality test.
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.