Number Theory

    

A Spectral Characterization of Primes Via Paley Graph Eigenvalues

Authors: F. F. Martinez Gamo

We present computational evidence for a novel spectral characterization of prime numbers through the Laplacian eigenvalues of Paley-type graphs. For integers n ≡ 1 (mod 4), we demonstrate that the second smallest Laplacian eigenvalue λu2082 of the graph constructed from quadratic residues modulo n satisfies λu2082(G) = (n - √n)/2 if and only if n is prime, with numerical precision limited only by floating-point accuracy (~10u207b¹u2075). Composite numbers exhibit substantial deviation from this formula, with gaps ranging from 3 to over 60 for n < 300. Statistical analysis over 29 primes and 30 composites shows a separation ratio exceeding 10¹u2075 between prime and composite gap magnitudes. This result establishes a connection between number-theoretic primality and graph spectral properties, with implications for understanding the algebraic structure of finite fields versus rings with zero divisors.

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[v1] 2025-11-22 01:10:10

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