用循环矩阵特征值判断素数，理论严谨且可确定性验证。
Primality Testing via Circulant Matrix Eigenvalue Structure: A Novel Approach Using Cyclotomic Field Theory
- 通过单位根构造循环矩阵，利用其最小多项式因式分解数判别素数。
- 整数n>2为素数当且仅当矩阵最小多项式在有理数域上有两个不可约因子。
- 适合对代数数论与素数判定感兴趣的数学及理论计算机研究者。
本文提出一种基于单位根构造的循环矩阵特征值结构的新型素性测试方法。证明对于大于2的整数n，当且仅当矩阵 $C_n = W_n + W_n^2$ 的最小多项式在 $/mathbb{Q}$ 上恰好有两个不可约因子时，n为素数。该刻画将分圆域理论与矩阵代数相结合，揭示了素数与合数在代数结构上的本质差异。研究展示了矩阵特征值模式能有效区分素数与合数，从而实现确定性素性检测。方法利用原根、伽罗瓦理论与分圆多项式因子分解之间的关系，通过广泛实验验证了该方法在不同整数范围内的有效性，并分析了其计算复杂度与现有素性测试的比较。该数学框架具有直观的视觉解释，有助于理解区分素数的代数结构。实验表明，该方法提供了具有代数基础性能特征的确定性替代方案。
原文摘要 · Abstract (English)
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.
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