Proth Prime Search

About the Proth Prime Search
The Proth Prime Search was done in collaboration with the Proth Search project, and PrimeGrid has completely taken over the search since 2012. This search looks for primes in the form of k*2^n+1 with the condition 2^n > k. These primes 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 with LLR, it is then tested on PrimeGrid's servers for divisibility.

Before 2010
PrimeGrid's 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. That was accomplished well ahead of schedule and the range has been increased 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.

After 2010
PPS has been split into four subprojects in different times after 2010-- PPS, PPSE, PPS-MEGA, and PPS-DIV.

PPS does the original search on odd 300<k<1200 to n=3.322M and is expected to reach that in 2022.

PPSE searches in the range of odd 1200<k<10000 to n=3.322M and is expected to reach that in around 15 years.

PPS-MEGA gets a bit complicated:

Before July 2019, PPS-MEGA searched odd 300<k<1200 to 3.322M<n<3.6M.

Since that range has been exahusted in July 2019, new work has been loaded for odd 1200<k<10000 in the same n range. This is expected to finish around 2023.

PPS-DIV is complicated too:

Starting in September 2019, k=19683 was searched up to n=4M first, ending in October 2019.

In September 2019, k=1323, 2187, 3125, 3267, 3375 were added and searched to n=3.322M in October 2019.

At n=3.6M in October 2019, odd k's 10<k<50 were added to the search.

At n=4M in October 2019, k=9 were added to the search.

At n=6M some time around May 2020, k=5, 7 will be added to the search.

Some time in 2022, 3<k<50 will reach n=9M and the project will end permanently.

All the above PPS projects count towards the PPS Badge.

For more information about Proth primes, please visit these links:


 * Proth Prime - The Prime Glossary
 * Proth Prime - Wolfram MathWorld
 * Proth Number - Wikipedia

About Proth Search

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:


 * Lucas-Lehmer-Riesel test (WIKI)
 * Download LLR by Jean Penné

(Edouard Lucas: 1842-1891, Derrick H. Lehmer: 1905-1991, Hans I. Riesel: 1929-2010).