Mathematics > Number Theory
arXiv:2606.22851 (math)
[Submitted on 22 Jun 2026]
Abstract:We describe several new algorithms for finding all prime numbers up to a given bound $N$, achieving the first ever speedup by a positive power of $\log N$ over the ancient sieve of Eratosthenes. The fastest version, which is not fully rigorous, runs in \[ N (\log \log N)^{1+o(1)} \] bit operations when analysed in the multitape Turing model. This improves on the best existing algorithms due to Pritchard (1981), Atkin--Bernstein (2004) and Sergeev (2016) by a factor of almost $\log N$. We also present a rigorous randomised (Las Vegas) variant that is slower by a factor of $(\log \log N)^{1+o(1)}$, and a rigorous deterministic variant that is slower by a factor of $(\log N)^{1/2+o(1)}$. The new algorithms make heavy use of fast polynomial arithmetic over finite fields, and also involve ideas from the theory of error-correcting codes.
| Comments: | 121 pages |
| Subjects: | Number Theory (math.NT); Data Structures and Algorithms (cs.DS); Symbolic Computation (cs.SC) |
| MSC classes: | 11Y16, 11Y11 (Primary) 11N36, 68W40, 68W30, 11T71 (Secondary) |
| ACM classes: | F.2.1; I.1.2; E.4 |
| Cite as: | arXiv:2606.22851 [math.NT] |
| (or arXiv:2606.22851v1 [math.NT] for this version) | |
| https://doi.org/10.48550/arXiv.2606.22851 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: David Harvey [view email]
[v1]
Mon, 22 Jun 2026 04:54:20 UTC (133 KB)
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