Summary: A computational investigation using spectral analysis of residue-circulant matrices has discovered that primes congruent to 1 mod 4 exhibit a vanishing "Paley gap"—the discrepancy between computed and theoretical eigenvalues reaches machine precision ( $10^{- 15}$ )). This phenomenon connects quadratic residue theory, optimal graph expansion properties (Ramanujan graphs), and Fast Fourier Transform methods, offering new computational perspectives on prime number structure. While not constituting a primality test, the discovery demonstrates how modern algorithmic frameworks can illuminate classical mathematical structures.

Research outputs:

Preprint: Spectral note: Paley gap via lambda_2 (residue circulants) on Zenodo
Code repository: TNFR-Python-Engine on GitHub
Accessible explanation: Medium article

The discovery

Recent work emerging from the TNFR computational framework has identified a remarkable spectral property of certain prime numbers. For primes $p \equiv 1 (mod 4),$ the algebraic connectivity (second-smallest Laplacian eigenvalue (\(\lambda_2\\))) of the associated Paley graph matches its theoretical prediction with extraordinary precision—to within floating-point rounding error.

This "Paley gap" (\(g(p) = |\lambda_2 - (p - \sqrt{p})/2|\\)) effectively vanishes for tested primes up to 2601, consistently achieving values around \(10^{-13}\\) to $10^{- 15}$ . The phenomenon has been verified across dozens of primes and appears robust.

The complete numerical results, methodology and reproducible code are available in the Zenodo preprint and the TNFR-Python-Engine repository.

Mathematical background: from quadratic residues to graph spectra

A quadratic residue modulo prime $p$ is an integer $a$ such that \(x^2 \equiv a \pmod{p}\\) has a solution. For example, modulo 13, the quadratic residues are {1, 3, 4, 9, 10, 12}—exactly half of the non-zero elements. These residues encode deep arithmetic information about the prime's algebraic structure.

The Legendre symbol \(\left(\frac{a}{p}\right)\\) formalizes this: it equals $+ 1$ if \(a\\) is a quadratic residue modulo $p$ , $- 1$ if not and $0$ if $p | a$ . A fundamental result states that for primes $p \equiv 1 (mod 4)$ , the element $- 1$ is itself a quadratic residue. This congruence condition ensures crucial symmetry properties in what fo...

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