A Spectral-Fractal Proof of the Erdos-Graham Conjecture
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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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2025
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| _version_ | 1866902280835432448 |
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| author | Maréchal, Thierry |
| author_facet | Maréchal, Thierry |
| contents | <p>We present a complete spectral-fractal proof of the Erdos-Graham Conjecture, which asserts that for any integer n > 1 and any finite partition of the prime numbers into disjoint subsets S1, S2, . . . , Sr , there exists at least one subset Si with Pp∈Si1p > n.</p> <p>Building on our universal spectral-fractal framework with characteristic fractal dimension √2 ≈1.414, we construct a family of self-adjoint Sturm-Liouville operators<br>HSi = − d2dx2 + Vfractal(x; Si) on the domain (0, ∞) that encode the reciprocal sum properties of prime subsets.</p> <p>We prove that the spectral density constraints, coupled with the divergence of the overall prime reciprocal sum, ensure that at least one subset must have an arbitrarily large reciprocal sum. This provides a novel analytical perspective on the distribution of prime numbers and the behavior of their reciprocal sums under partitioning.</p> <p>Our theoretical results are validated through high-performance computing tests on prime partitions up to p = 106.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15116797 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Spectral-Fractal Proof of the Erdos-Graham Conjecture Maréchal, Thierry Mathematics <p>We present a complete spectral-fractal proof of the Erdos-Graham Conjecture, which asserts that for any integer n > 1 and any finite partition of the prime numbers into disjoint subsets S1, S2, . . . , Sr , there exists at least one subset Si with Pp∈Si1p > n.</p> <p>Building on our universal spectral-fractal framework with characteristic fractal dimension √2 ≈1.414, we construct a family of self-adjoint Sturm-Liouville operators<br>HSi = − d2dx2 + Vfractal(x; Si) on the domain (0, ∞) that encode the reciprocal sum properties of prime subsets.</p> <p>We prove that the spectral density constraints, coupled with the divergence of the overall prime reciprocal sum, ensure that at least one subset must have an arbitrarily large reciprocal sum. This provides a novel analytical perspective on the distribution of prime numbers and the behavior of their reciprocal sums under partitioning.</p> <p>Our theoretical results are validated through high-performance computing tests on prime partitions up to p = 106.</p> |
| title | A Spectral-Fractal Proof of the Erdos-Graham Conjecture |
| topic | Mathematics |
| url | https://doi.org/10.5281/zenodo.15116797 |