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JimB
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Message 25709 - Posted: 20 Aug 2010 | 14:51:53 UTC
Last modified: 5 May 2013 | 12:49:20 UTC

About the Proth Prime Search

The Proth Prime Search is done in collaboration with the Proth Search project. This search looks for primes in the form of k*2^n+1. With the condition 2^n > k, these are often called Proth primes. This project also has the added bonus of possibly finding factors of "classical" Fermat numbers or Generalized Fermat numbers. As this requires PrimeFormGW (PFGW) (a primality-testing program), once PrimeGrid finds a prime, it is then tested on PrimeGrid's servers for divisibility.

Our initial goal was to double check all previous work up to n=500K for odd k<1200 and to fill in any gaps that were missed. We have accomplished that now and have increased it to n=800K. PG LLRNet searched up to n=200,000 and found several missed primes in previously searched ranges. Although primes that small did not make it into the Top 5000 Primes database, the work was still important as it may have led to new factors for "classical" Fermat numbers or Generalized Fermat numbers. While there are many GFN factors, currently there are only about 275 "classical" Fermat number factors known. Current primes found in PPS definitely make it into the Top 5000 Primes database.

Once the 800K goal is reached, we may head to 1M before turning our focus to smaller k values and higher n values. For example, k<300 complete to n=2M, k<600 complete to n=1.5M and so on.

The Proth Search project was established in 1998 by Ray Ballinger and Wilfrid Keller to coordinate a distributed effort to find Proth primes (primes of the form k*2^n+1) for k < 300. Ray was interested in finding primes while Wilfrid was interested in finding divisors of Fermat number. Since that time it has expanded to include k < 1200. Mark Rodenkirch (aka rogue) has been helping Ray keep the website up to date for the past few years.

Early in 2008, PrimeGrid and Proth Search teamed up to provide a software managed distributed effort to the search. Although it might appear that PrimeGrid is duplicating some of the Proth Search effort by re-doing some ranges, few ranges on Proth Search were ever double-checked. This has resulted in PrimeGrid finding primes that were missed by previous searchers. By the end of 2008, all new primes found by PrimeGrid were eligible for inclusion in Chris Caldwell's Prime Pages Top 5000. Sometime in 2009, over 90% of the tests handed out by PrimeGrid were numbers that have never been tested. For 2010, we hope to complete our reservation to 800K and extend it to 1M.

PrimeGrid intends to continue the search indefinitely for Proth primes.

What is LLR?

The Lucas-Lehmer-Riesel (LLR) test is a primality test for numbers of the form N = k*2^n − 1, with 2^n > k. Also, LLR is a program developed by Jean Penne that can run the LLR-tests. It includes the Proth test to perform +1 tests and PRP to test non base 2 numbers. See also:

(Edouard Lucas: 1842-1891, Derrick H. Lehmer: 1905-1991, Hans Riesel: born 1929).

Message boards : Proth Prime Search : Proth Prime Search