Elliptic Curve Primality Proof

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The Prime Pages

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page.This page is about one of those forms.

Definitions and Notes

This page lists record primes that were first proven prime by the elliptic curve primality proving algorithm. It is shown here as a convenience for those watching the heated contest between the chief ECPP programmers. Originally these were François Morain (who first set a titanic prime record for proving primality via ECPP) and Marcel Martin (who wrote a version called Primo for Windows machines). In 2003, J. Franke, T. Kleinjung and T. Wirth greatly increased the size of numbers that could be handled with a new program of their own. Morain has worked with this trio and they have both improved their programs [FKMW2003]. Martin's Primo is by far the easiest of these programs to set up and use. There seems to be some question which is fastest on a single CPU.

ECPP has replaced the groups of order n-1 and n+1 used in the classical test with a far larger range of group sizes (see our page on elliptic curve primality proving). The idea is that we can keep switching elliptic curves until we find one we can "factor". This improvement comes at the cost of having to do a great deal of work to find the actual size of these groups--but works for all numbers, not just those with very special forms.

About 1986 S. Goldwasser & J. Kilian [GK86] and A. O. L. Atkin [Atkin86] introduced elliptic curve primality proving methods. Atkin's method, ECPP, was implemented by a number of mathematicians, including Atkin & Morain [AM93]. Heuristically, ECPP is O((log n)^5+eps) (with fast multiplication techniques) for some eps > 0 [LL90]. It has been proven to be polynomial time for almost all choices of inputs. A version attributed to J. O. Shallit is O((log n)^4+eps). Franke, Kleinjung and Wirth combined with Morain to improve their respective programs (both now use Shallit's changes), creating what they "fastECPP" [FKMW2003].

The editors expect this page should remain our only Top Twenty Page dedicated to a proof method rather than a form of prime. Note that "fastECPP" is simply a name--their use of the adjective 'fast' should not be construed as a comparison to programs by other authors (which may also follow Shallit's approach).

Record Primes of this Type

rank prime digits who when comment

1 p(1289844341) 40000 c84 Feb 2020 Partitions, ECPP
2 2¹¹⁶²²⁴ - 15905 34987 c87 Nov 2017 ECPP
3 (14665 · 10³⁴¹¹⁰ - 56641)/9999 34111 c89 Mar 2018 ECPP, Palindrome
4 "10000000000000000000...(34053 other digits)...00000000000000532669" 34093 c84 Nov 2016 ECPP
5 V(148091) 30950 c81 Sep 2015 Lucas number, ECPP
6 U(148091) 30949 x49 Sep 2021 Fibonacci number, ECPP
7 " - τ(331²¹²⁸)" 29492 c80 Sep 2015 ECPP
8 V(140057) 29271 c76 Dec 2014 Lucas number, ECPP
9 (2⁹⁵³⁶⁹ + 1)/3 28709 x49 Aug 2021 Generalized Lucas number, Wagstaff, ECPP
10 - 30 · Bern(10264)/(1040513 · 252354668864651) 28506 c94 May 2021 Irregular, ECPP
11 (10²⁷⁶⁶⁹ + 7)/8313493832818655929448065598763458531111 27630 c96 Feb 2021 ECPP
12 U(130021) 27173 x48 May 2021 Fibonacci number, ECPP
13 546351925018076058 · Bern(9702)/129255048976106804786904258880518941 26709 c77 Jan 2021 Irregular, ECPP
14 "τ(157²²⁰⁶)" 26643 FE1 Apr 2011 ECPP
15 10²⁵³³³ - 2 · 10⁵¹⁸² - 3 25333 c95 Aug 2020 ECPP
16 Phi(12345, 7176)/31531760245313526865033921 25331 c54 Jun 2017 ECPP
17 (2⁸⁴²¹¹ - 1)/1347377 / 31358793176711980763958121 / 3314641676042347824169591561 25291 c95 Sep 2020 Mersenne cofactor, ECPP
18 (2⁸³³³⁹ + 1)/3 25088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff
19 6753⁵¹²² + 5122⁶⁷⁵³ 25050 FE1 Oct 2010 ECPP
20 (2⁸²⁹³⁹ - 1)/883323903012540278033571819073 24938 c84 Feb 2021 Mersenne cofactor, ECPP

ECPP From the Prime Glossary
Short introduction to ECPP

References

AM93
A. O. L. Atkin and F. Morain, "Elliptic curves and primality proving," Math. Comp., 61:203 (July 1993) 29--68. MR 93m:11136
Atkin86
A. O. L. Atkin, "Lecture notes of a conference," Boulder Colorado, (August 1986) Manuscript. [See also [AM93].]
FKMW2003
Franke, J., Kleinjung, T., Morain, F. and Wirth, T., Proving the primality of very large numbers with fastECPP. In "Algorithmic number theory," Lecture Notes in Comput. Sci. Vol, 3076, Springer, Berlin, 2004. pp. 194--207, (http://dx.doi.org/10.1007/978-3-540-24847-7_14) MR 2137354
GK1999
S. Goldwasser and J. Kilian, "Primality testing using elliptic curves," J. ACM, 46:4 (1999) 450--472. MR 2002e:11182
GK86
S. Goldwasser and J. Kilian, Almost all primes can be quickly certified. In "STOC'86, Proceedings of the 18th Annual ACM Symposium on the Theory of Computing (Berkeley, CA, 1986)," ACM, May 1986. New York, NY, pp. 316--329,
LL90
Lenstra, Jr., A. K. and Lenstra, Jr., H. W., Algorithms in number theory. In "Handbook of Theoretical Computer Science, Vol A: Algorithms and Complexity," The MIT Press, Amsterdam and New York, 1990. pp. 673-715, MR 1 127 178
Morain98
F. Morain, Primality proving using elliptic curves: an update. In "Algorithmic Number Theory, Third International Symposium, ANTS-III," J. P. Buhler editor, Lecture Notes in Comput. Sci. Vol, 1423, Springer-Verlag, June 1998. pp. 111--127, MR 2000i:11190