Divisors in Residue Classes Revisited
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909338710310912 |
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| author | Hales, Jonathon |
| author_facet | Hales, Jonathon |
| contents | In 1984, H. W. Lenstra described an algorithm finding divisors of $N$ congruent to $r \mod S$. When $S^3 > N$, this algorithm runs in polynomial time and hence factors $N$ in time $N^{1/3+o(1)}$. Lenstra's algorithm relies on a sign change in a constructed sequence and so cannot be adapted directly to larger euclidean number rings. We present a new method that generalizes to a larger class of euclidean rings and the polynomial ring $\mathbb{Z}[x]$. The algorithm is implemented and timed confirming its polynomial run time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05030 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Divisors in Residue Classes Revisited Hales, Jonathon Number Theory 11Y16 (Primary), 11Y40, 11Y05 (Secondary) In 1984, H. W. Lenstra described an algorithm finding divisors of $N$ congruent to $r \mod S$. When $S^3 > N$, this algorithm runs in polynomial time and hence factors $N$ in time $N^{1/3+o(1)}$. Lenstra's algorithm relies on a sign change in a constructed sequence and so cannot be adapted directly to larger euclidean number rings. We present a new method that generalizes to a larger class of euclidean rings and the polynomial ring $\mathbb{Z}[x]$. The algorithm is implemented and timed confirming its polynomial run time. |
| title | Divisors in Residue Classes Revisited |
| topic | Number Theory 11Y16 (Primary), 11Y40, 11Y05 (Secondary) |
| url | https://arxiv.org/abs/2410.05030 |