# Elliptic Curve Primality Proof

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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. Comments and suggestions requested.

### 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

rankprime digitswhowhencomment
12116224 - 15905 34987 c87 Nov 2017 ECPP
2(14665 · 1034110 - 56641)/9999 34111 c89 Mar 2018 ECPP, Palindrome
3"10000000000000000000...(34053 other digits)...00000000000000532669" 34093 c84 Nov 2016 ECPP
4V(148091) 30950 c81 Sep 2015 Lucas number, ECPP
5" - τ(3312128)" 29492 c80 Sep 2015 ECPP
6V(140057) 29271 c76 Dec 2014 Lucas number, ECPP
7"τ(1572206)" 26643 FE1 Apr 2011 ECPP
8Phi(12345, 7176)/31531760245313526865033921 25331 c54 Jun 2017 ECPP
9(283339 + 1)/3 25088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff
1067535122 + 51226753 25050 FE1 Oct 2010 ECPP
11floor((3 / 2)137752) + 13566 24257 c35 Mar 2015 ECPP
12" - τ(6911522)" 23770 c65 Mar 2014 ECPP
13"τ(2571698)" 22506 c72 Apr 2014 ECPP
141022250 + 57913 22251 c35 May 2014 ECPP
15273845 + 14717 22230 c61 Dec 2013 ECPP
16273360 + 10711 22084 c61 Sep 2014 ECPP
17U(104911) 21925 c82 Oct 2015 Fibonacci number, ECPP
1820170108 · 1021700 + 39 21708 c83 Nov 2016 ECPP
19((((((25210088873 + 80)3 + 12)3 + 450)3 + 894)3 + 3636)3 + 70756)3 + 97220 20562 FE1 Jun 2006 ECPP, Mills' prime
20"τ(412296)" 20367 c91 Apr 2018 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, New York, NY, May 1986.  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, 1990.  Amsterdam and New York, 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