| _version_ | 1866901930932961280 |
|---|---|
| author | Amirov, Zakir |
| author_facet | Amirov, Zakir |
| contents | <p>We show that for all sufficiently large <span><span>N</span></span> there exists an integer <span><span>n∈[N,6N^2]</span></span> such that <span><span><span>6n≢±1 (mod p) </span></span></span>for every prime <span><span>5≤p≤6N+1</span></span>. The proof uses explicit Type I/II sieve estimates to show that the set of integers failing this condition has size <span><span>O(X^{<span><span><span><span><span><span><span><span><span>1<span>−</span><span>δ</span></span></span></span></span></span></span></span></span></span>})</span></span> with <span><span>X=6N^2</span></span>. As a result, such integers occur infinitely often.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17929262 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Explicit type I/II proof of integers avoiding two forbidden residues in long intervals Amirov, Zakir Sieve methods analytic number theory arithmetic progressions prime divisors <p>We show that for all sufficiently large <span><span>N</span></span> there exists an integer <span><span>n∈[N,6N^2]</span></span> such that <span><span><span>6n≢±1 (mod p) </span></span></span>for every prime <span><span>5≤p≤6N+1</span></span>. The proof uses explicit Type I/II sieve estimates to show that the set of integers failing this condition has size <span><span>O(X^{<span><span><span><span><span><span><span><span><span>1<span>−</span><span>δ</span></span></span></span></span></span></span></span></span></span>})</span></span> with <span><span>X=6N^2</span></span>. As a result, such integers occur infinitely often.</p> |
| title | Explicit type I/II proof of integers avoiding two forbidden residues in long intervals |
| topic | Sieve methods analytic number theory arithmetic progressions prime divisors |
| url | https://doi.org/10.5281/zenodo.17929262 |