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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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Zenodo
2025
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| Acceso en línea: | https://doi.org/10.5281/zenodo.15347108 |
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| _version_ | 1866902021515247616 |
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| author | Fathi, Kevin |
| author_facet | Fathi, Kevin |
| contents | <p>We prove that there are only finitely many prime values in the base-10 repunit<br>sequence<br>Rn = (10n − 1)/9 .<br>Our proof relies purely on structural properties of repunits, combining cyclotomic<br>factorization, growth of primitive prime divisors via Zsigmondy’s Theorem, and sym-<br>bolic entropy compression arguments. We show that as n → ∞, the redundancy and<br>multiplicative structure of Rn necessarily force compositeness beyond a finite bound,<br>independent of computational residue checks.</p> <p><br>By analyzing the entropy structure of repeated-digit numbers and the accumulation<br>of distinct prime divisors, we demonstrate that repunit primes must be confined to<br>finitely many small indices. The result is a complete classical proof confirming the<br>longstanding conjecture that only finitely many base-10 repunit numbers are prime.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15347108 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | On the Finiteness of Base-10 Repunit Primes: A Structural Proof via Cyclotomic Factorization and Divisor Growth Fathi, Kevin Repunits Repunit primes Bit-length growth Logarithmic density Prime rarity Number theory <p>We prove that there are only finitely many prime values in the base-10 repunit<br>sequence<br>Rn = (10n − 1)/9 .<br>Our proof relies purely on structural properties of repunits, combining cyclotomic<br>factorization, growth of primitive prime divisors via Zsigmondy’s Theorem, and sym-<br>bolic entropy compression arguments. We show that as n → ∞, the redundancy and<br>multiplicative structure of Rn necessarily force compositeness beyond a finite bound,<br>independent of computational residue checks.</p> <p><br>By analyzing the entropy structure of repeated-digit numbers and the accumulation<br>of distinct prime divisors, we demonstrate that repunit primes must be confined to<br>finitely many small indices. The result is a complete classical proof confirming the<br>longstanding conjecture that only finitely many base-10 repunit numbers are prime.</p> |
| title | On the Finiteness of Base-10 Repunit Primes: A Structural Proof via Cyclotomic Factorization and Divisor Growth |
| topic | Repunits Repunit primes Bit-length growth Logarithmic density Prime rarity Number theory |
| url | https://doi.org/10.5281/zenodo.15347108 |