EMsieve

A titan, as defined by Samuel Yates, is anyone who has found a titanic prime. This page provides data on those that have found these primes. The data below only reflects on the primes currently on the list. (Many of the terms that are used here are explained on another page.)

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Proof-code(s):	L3839, p294, L4026, p379, L4142, L4506, L4561, L5123
E-mail address:	(e-mail address unpublished)
Web page:	http://sourceforge.net/projects/emsieve/
Username:	EMsieve	(entry created on 03/29/14)
Database id:	4141	(entry last modified on 07/31/20)
Program Does *:	sieve
Active primes:	on current list: 69, rank by number 12
Total primes:	number ever on any list: 124
Production score:	for current list 52 (normalized: 4552), total 52.8155, rank by score 13
Largest prime:	Phi(3, - 123447⁵²⁴²⁸⁸) ‏(‎5338805 digits) via code L4561 on 02/23/17
Most recent:	2^15317227 + 2^7658614 + 1 ‏(‎4610945 digits) via code L5123 on 07/31/20
Entrance Rank:	mean 540.71 (minimum 12, maximum 1760)

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This code corresponds to both a simple sieve/prefactor program for the so-called Eisenstein-Mersenne Primes: 3^p +- 3^((p + 1)/2) + 1, and a special modified variant of LLR (with due credit for the original framework code to Jean Penne, and to George Woltman for GWNUM).

See http://oeis.org/A066408, A125739, and [1] for a good introduction. Some easily established properties are: p must be prime; sign is minus for p=+-1 (mod 12), plus otherwise; composites only have factors of form 6kp+1 (integer k).

After sieving, the Berrizbeitia-Iskra or the Proth test can be run; this is best implemented with FFT mod (3^3p+1) using GWNUM library. A sample implementation (a patch to the LLR program) is available from Batalov.

Also, this code is extended to an accessory GPU-assisted sieve for pre-factoring both Eisenstein-Mersenne and Gaussian-Mersenne candidates. This CUDA program is adapted from well-known mfaktc [2].

P.Berrizbeitia, B.Iskra, 2010; http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.189.311
OEIS: http://oeis.org/
http://www.mersennewiki.org/index.php/Mfaktc

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Surname: EMsieve (used for alphabetizing and in codes)
Unverified primes are omitted from counts and lists until verification completed.

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