Hana's conjecture
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2025
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| author | Elgamala, Hana Abdelrahman |
| author_facet | Elgamala, Hana Abdelrahman |
| contents | <p>This research proposes a structural conjecture regarding the prime divisors of numbers of the form <span><span>p+4</span></span>, where <span><span>pp</span><span><span><span>p</span></span></span></span> is prime. While cousin primes <span><span>(p,p+4)</span></span> are well-known in number theory, the compositeness of <span><span>p+4</span></span> reveals a striking modular pattern: if <span><span>p+4 </span></span>is composite, it always has a prime divisor congruent to 1, 3, or 7 modulo 10, or one of the base primes 3, 7, or 11. The paper provides computational evidence up to <span><span>10^6</span></span> and discusses heuristic and partial theoretical motivations, connecting the conjecture to modular arithmetic, additive structures, and covering systems. The work also suggests natural generalizations for other shifted primes and raises open problems for further investigation.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17937427 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Hana's conjecture Elgamala, Hana Abdelrahman Prime numbers cousin primes number theory conjecture <p>This research proposes a structural conjecture regarding the prime divisors of numbers of the form <span><span>p+4</span></span>, where <span><span>pp</span><span><span><span>p</span></span></span></span> is prime. While cousin primes <span><span>(p,p+4)</span></span> are well-known in number theory, the compositeness of <span><span>p+4</span></span> reveals a striking modular pattern: if <span><span>p+4 </span></span>is composite, it always has a prime divisor congruent to 1, 3, or 7 modulo 10, or one of the base primes 3, 7, or 11. The paper provides computational evidence up to <span><span>10^6</span></span> and discusses heuristic and partial theoretical motivations, connecting the conjecture to modular arithmetic, additive structures, and covering systems. The work also suggests natural generalizations for other shifted primes and raises open problems for further investigation.</p> |
| title | Hana's conjecture |
| topic | Prime numbers cousin primes number theory conjecture |
| url | https://doi.org/10.5281/zenodo.17937427 |