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JimBVolunteer moderator Project developer Project scientist Send message
Joined: 4 Aug 11 Posts: 623 ID: 107307 Credit: 424,891,226 RAC: 2,373,897

PrimeGrid's PSA PRPNet Projects
The projects below are all available through PrimeGrid's PRPNet servers. Clicking on their names will take you their overview post. Listed under each project is its server port(s).
To learn more about PRPNet and find the latest release, please see the Welcome to PRPNet thread.
Detailed Daily User/Team Stats:
Detailed daily user stats courtesy of Sysadm@Nbg
Today's Prime Finders courtesy of Sysadm@Nbg
Current PrimeGrid PRPNet Projects and Servers
Retired Ports
port 12007: PPSEhigh (Now being run in BOINC): Stats
port 13000: 5oB (The Dual Sierpinski Problem (COMPLETED AND CLOSED): Stats
port 9000: PPSE n>500K User Stats
port 10000: PPSE n<500K
port 12000: Proth Prime Search (COMPLETED AND CLOSED)
port 12000: Sophie Germain Prime Search (SUSPENDED)
port 14000: The Riesel Problem Double Check (COMPLETED AND CLOSED)
port 7171: Sierpinski/Riesel Base 5 Project (MOVED TO BOINC)
port 13005: Extended Sierpinski Problem (MOVED TO BOINC)
port 12010: Mega Prime Search (MOVED TO BOINC)
port 12005: GFN32768 (MOVED TO BOINC)
port 12003: GFN65536 (MOVED TO BOINC)
port 11002: GFN262144 (MOVED TO BOINC)
port 11001: GFN524288 (MOVED TO BOINC)
port 12004: Generalized Cullen/Woodall (MOVED TO BOINC)  

JimBVolunteer moderator Project developer Project scientist Send message
Joined: 4 Aug 11 Posts: 623 ID: 107307 Credit: 424,891,226 RAC: 2,373,897

Completed!
Welcome 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 primalitytesting 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.
Additional Proth prime testing can be found in PrimeGrid's PRPNet's Proth Prime Search Extended (PPSE) project. This project extends the search range to 1200<k<10000 for n<2M. Current ports available are:
PRPNet Ports
Detailed daily user stats courtesy of Sysadm@Nbg
Today's Prime Finders courtesy of Sysadm@Nbg
port 12000: PPSE n<500K Server  User  Pending Tests  Primes  All
port 12007: PPSE n>700K Server : User : Pending Tests  Primes  All SUSPENDED (Now being run in BOINC.)
For more information about "Proth" primes, please visit these links:
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 redoing some ranges, few ranges on Proth Search were ever doublechecked. 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 LucasLehmerRiesel (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 LLRtests. It includes the Proth test to perform +1 tests and PRP to test non base 2 numbers. See also:
(Edouard Lucas: 18421891, Derrick H. Lehmer: 19051991, Hans Riesel: born 1929).
 

JimBVolunteer moderator Project developer Project scientist Send message
Joined: 4 Aug 11 Posts: 623 ID: 107307 Credit: 424,891,226 RAC: 2,373,897

Welcome to the Sophie Germain Prime Search
The Sophie Germain Prime search honors MarieSophie Germain, an extraordinary "French mathematician who made important contributions to the fields of differential geometry and number theory, and to the study of Fermat's Last Theorem." (Wiki)
A prime number p is called a Sophie Germain prime if 2p + 1 is also prime. For example, 5 is a Sophie Germain prime because it is prime and 2 × 5 + 1 = 11, is also prime.
We'll be searching the form k*2^n1. If it is prime, then we'll check k*2^n+1, k*2^(n1)1, & k*2^(n+1)1. We are able to do this because a quad sieve was performed for this search. This sieve ensured that k*2^n1, k*2^n+1, k*2^(n1)1, & k*2^(n+1)1 did not have any small prime divisors. The opportunity to find SG's and Twins in the same sieve file is appealing. However, we "expect" to find a Sophie Germain prime first.
This quad sieve was prepared quite some time ago; so it was readily available. Here are some stats for the search:
SUSPENDED (Now being run in BOINC.)
k range: 1<k<41T
n=666666 (actual 666666666685)
sieve depth: p=200T
candidates remaining: 0
Range Stats
Probability of one or more significant pair = 80.1%
Probability of one or more SG = 66.7%
Probability of one or more Twin = 42.3%
Approximate WU length:
Athlon64 2.1Ghz  ~2000 secs (~33.3 minutes)
C2D 2.1 Ghz  ~1015 secs (~16.9 minutes) per core
C2Q 2.4 GHz  ~880 secs (~14.7 minutes) per core
C2Q 3.2 GHz  ~600 secs (~10.0 minutes) per core
For more information about Sophie Germain primes, please visit these links:
http://primes.utm.edu/glossary/page.php?sort=SophieGermainPrime
http://mathworld.wolfram.com/SophieGermainPrime.html
http://en.wikipedia.org/wiki/Sophie_Germain_prime
For more infomation about MarieSophie Germain, please visit these links:
http://en.wikipedia.org/wiki/Sophie_Germain
http://www.pbs.org/wgbh/nova/proof/germain.html  

JimBVolunteer moderator Project developer Project scientist Send message
Joined: 4 Aug 11 Posts: 623 ID: 107307 Credit: 424,891,226 RAC: 2,373,897

Welcome to the Primorial Prime Search
In order to define a primorial prime, we must first define primorial. The primorial pn# is defined as the product of the first n primes. For example,
p5# = 2*3*5*7*11 = 2310
This can also be notated as p#, the product of all primes less than or equal to prime p. The above p5# would be 11#, the product of all primes less than or equal to prime 11.
Primorial primes are prime numbers of the form p#+/1. Using the example above, we would look to see if 11#+1 and 11#1 are prime.
11# = 2*3*5*7*11 = 2310
11#+1 = 2311 is prime
11#1 = 2309 is prime
Therefore, 11#+1 and 11#1 are both primorial primes. Using another example, 7#:
7# = 2*3*5*7 = 210
7#+1 = 211 is prime
7#1 = 209 is not prime
Therefore, only 7#+1 is a primorial prime.
To date, the largest known primorial prime is 1098133#1 with 476311 digits, found in PrimeGrid's PRPNet on 28 Feb 2012 by James P. Burt of the Cayman Islands. 1098133#1 = 2*3*5*7*...*1098101*1098109*1098121*1098133  1
p#+1 is prime for primes p=2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439 and 392113 (169966 digits).
p#1 is prime for primes p=3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301 and 1098133 (476311 digits).
A list of the top 20 primorial primes can be found at The Prime Pages: The Top 20
To participate in the search, add the following server/port to your prpclient.ini file (adjust project share accordingly):
server=PRS:0:1:prpnet.primegrid.com:12008
Server Stats
Mark Rodenkirch's (in collaboration with Geoff Reynolds) psieve program will be used to sieve and Chris Nash and Jim Fougeron's PGFW program will be used to primality test.
Additional information can be found here:
Primorial prime  The Prime Glossary at the Prime Pages
Primorial prime  Wikipedia
Primorial prime  Wolfram MathWorld

In 2005, a previous effort, "Coordinated Search for Primorial Primes", reached n=637K for p# + 1 and n=650K for p#  1. Up to those limits, PrimeGrid's Primorial Prime Search will be a doublecheck.
However, there were open reservations up to the following:
n<750K for p# + 1
n<730K for p#  1 It is unknown how much of these open reservations, if any, were completed.

To participate in the sieving effort, see here:
Primorial Prime Search Sieving  

JimBVolunteer moderator Project developer Project scientist Send message
Joined: 4 Aug 11 Posts: 623 ID: 107307 Credit: 424,891,226 RAC: 2,373,897

Welcome to the Factorial Prime Search
In order to define a factorial prime, we must first define factorial. The factorial of a positive integer (denoted as n!) is the product of all the positive integers less than or equal to it. For example,
4! = 4*3*2*1 = 24
6! = 6*5*4*3*2*1 = 720
Factorial primes are prime numbers of the form n!+/1. Using the example above, we would look to see if 4!+1 and 4!1 are prime.
4! = 4*3*2*1 = 24
4!+1 = 25 is not prime
4!1 = 23 is prime
Therefore, 4!1 is a factorial prime.
To date, the largest known factorial prime is 208003!1 with 1015843 digits, found on 25 July 2016 by Sou Fukui.
n!+1 is prime for n=1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059 (507082 digits) and 150209!+1 (712355 digits).
n!1 is prime for n=3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040 (471794 digits), 147855 (700176 digits) and 208003.
A list of the top 20 factorial primes can be found at The Prime Pages: The Top 20
To participate in the search, add the following server/port to your prpclient.ini file (adjust project share accordingly):
server=FPS:100:1:prpnet.primegrid.com:12002
Stats
Best of Luck!
Mark Rodenkirch's (in collaboration with Geoff Reynolds) fsieve program will be used to sieve and Chris Nash and Jim Fougeron's PGFW program will be used to primality test. EDIT: fsieve has been combined by Geoff Reynolds into a single program with psieve...thus fpsieve is what we'll use.
Additional information can be found here:
Factorial prime  The Prime Glossary at the Prime Pages
Factorial prime  Wikipedia
Factorial prime  Wolfram MathWorld

A previous ongoing effort can be found here: Factorial Prime Search. As of 2009, n!1 has been tested for n<37000 and n!+1 for n<35500 with additional noncontinuous ranges to n=71000. For the completed work, PrimeGrid's Factorial Prime Search will be a doublecheck. The ongoing effort is testing up to n=100,000. Therefore, once PrimeGrid has double checked all the completed work, the search will continue at n>100,000. PrimeGrid's current sieve file is for n<1M.

To participate in the sieving effort, see here:
 Factorial Prime Search Sieving  

JimBVolunteer moderator Project developer Project scientist Send message
Joined: 4 Aug 11 Posts: 623 ID: 107307 Credit: 424,891,226 RAC: 2,373,897

Welcome to the 27121 Prime Search
PrimeGrid and 12121 Search are now collaborating in an effort to find a Mega Prime of the form 121*2^n1. This is very similar to the 321 Prime Search at k=3.
12121 Search was established on 5/24/2004 to search for large primes of the form 121*2^n1. Later, they added k=27 to their search. Their "short term" goal is to search for n up to 10M. For more information, please visit their site: 12121 Search.
As we did with 321, we are going to test the +1 form of these k's in parallel as well. Therefore, the effort will consist of the following:
121*2^n+1, 121*2^n1
27*2^n+1, 27*2^n1
for n<10M
First order of business is to establish a coordinated sieving effort to bring all forms to the same depth. Here's the current status:
COMPLETED wrote: 121 1 at p=290T for n<10M
121 +1 at p~450T for n<5M and p=1T for 5M<n<10M
27 1 at p=130T for n<10M
27 +1 at p~450T for n<5M and p=1T for 5M<n<10M
We need two efforts but we'll only focus on one right now:
1. Primary focus  27 1 to bring it up to p=290T then combine with 121 to take to 450T
2. 121 & 27 +1 for 5M<n<10M to bring it up to 450T.
Current progress can be found in the manual sieve reservations. To participate, please see 12121 Sieving Reservation.
Second order of business is to primality test 27 1, 27 +1, and 121 +1 up to n~3.7M to combine it with 121 1. Therefore, candidates will be available for testing in PrimeGrid's PRPNet. If you are interested in participating, please see the PRPNet thread. Port 5000 is currently loaded with 27 & 121 +1 for 600K<n<1.7M.
Again, welcome to 27121 Prime Search and the chance of finding another Mega Prime!!! :)
____________
 

JimBVolunteer moderator Project developer Project scientist Send message
Joined: 4 Aug 11 Posts: 623 ID: 107307 Credit: 424,891,226 RAC: 2,373,897

Completed and closed!
Welcome to the dual Sierpinski Problem
(Otherwise known as the "Five or Bust" Project)
NOTE: This is solely a double check effort.
The "Five or Bust" Project is attempting to solve the dual Sierpinski problem. Just as Seventeen or Bust is attempting to prove that 78557 is the smallest positive odd integer k such that k*2^n + 1 is always composite, 5oB is attempting to prove that 78557 is the smallest positive odd integer such that k + 2^n is always composite.
On 9 Feb 2011, the last remaining candidate was determined a probable prime. Therefore, the project has successfully concluded its active search. Other than proving that the probable primes really are prime (a monumental effort), all that remains is to confirm the found PRP's are the smallest PRP for each k. This requires a double check of all candidates below the found PRP's. Once this is complete, the Sloane sequence A067760 can be established.
PrimeGrid will be coordinating this double check effort. This is an excellent opportunity to help facilitate the ending to a successful project. To participate in the search, add the following server/port to your prpclient.ini file (adjust project share accordingly):
server=5OB:100:1:pgllr.mine.nu:13000
Stats
NOTE: For those new to PRPNet, please see the Welcome to PRPNet post.
The double check will be for the following k and n ranges:
k=2131 for 1250056<n<4582936
k=40291 for 2282200<n<9091912
k=41693 for 2000327<n<5146239
History
There was an ongoing search on the dual Sierpinski problem in 2002, and Payam Samidoost maintained a website on the status entitled The dual Sierpinski problem search. Phil Moore began searching the 8 unresolved sequences in August 2007, and discovered 3 additional large probable primes.
In October of 2008, Phil started "Five or Bust" to search the 5 remaining values. Please see this post for more information about the "Five or Bust" project and the dual Sierpinski problem.
In January 2011, a collaboration was established with PrimeGrid. In February, PRPNet was updated and testing began in the PST forum on double check work. However, before the joint effort was officially announced, the last candidate was eliminated.
____________
 

JimBVolunteer moderator Project developer Project scientist Send message
Joined: 4 Aug 11 Posts: 623 ID: 107307 Credit: 424,891,226 RAC: 2,373,897

Welcome to the Wieferich Prime Search
A prime p is a Wieferich prime if p^2 divides 2^(p1)  1. They are named after Arthur Wieferich who in 1909 proved that if the first case of Fermat’s last theorem is false for the exponent p, then p satisfies the criteria a^(p1) = 1 (mod p^2) for a=2.
Notice the similarity in the expression p^2 divides 2^(p1)  1 to the special case of Fermat's little theorem p divides 2^(p1)  1.
Despite a number of extensive searches, the only known Wieferich primes to date are 1093 and 3511. The rarity of these primes has lead to an interest in "Near" Wieferich primes. They are defined as special instances (with small A) of 2^((p1)/2) (mod p^2). Here's a list of "near" Wieferich primes (coming soon) with p > 200.
Search History
The search for Wieferich and Near Wieferich primes has been ongoing for over 70 years. Here's a summary of progress:
Search limit Author Year
16000 Beeger 1940
50000 Froberg unknown
100000 Kravitz 1960
200183 Pearson 1964
500000 Riesel 1964
3e7 Froberg 1968
3e9 Brillhart, Tonascia, and Weinberger 1971
6e9 Lehmer 1981
6.1e10 Clark 1996
4e12 Crandall, Dilcher, and Pomerance 1997
4.6e13 Brown and McIntosh 2001
2e14 Crump 2002
1.25e15 Knauer and Richstein 2005
3e15 Carlisle, Crandall, and Rodenkirch 2006
6.7e15 Dorais and Klyve 2011
10e15 PrimeGrid 20120113
14e15 PrimeGrid 20120414
14e16 PrimeGrid 20140811
Although the search for Wieferich primes reached an upper bound of 6.7e15, PrimeGrid's search will begin at 3e15. This gives us a large overlap to be able to compare results with the previous search. We do not expect to find any Wieferich primes between 3e15 and 6.7e15, but we do expect some "near" Wieferich primes.
Classical Definition of nearness
From Wiki: "A prime p satisfying the congruence 2^((p−1)/2) ≡ ±1 + Ap (mod p^2) with small A is commonly called a nearWieferich prime." Therefore, we are going to classify finds as follows:
Wieferich prime: A = 0
A <= 10
A <= 100
A <= 1000
For a list of nearWieferich primes meeting the above criteria see here:
Wieferich  sortable
Wieferich  pdf How to Participate
To participate in the search, add the following server/port to your prpclient.ini file (adjust project share accordingly):
server=WIEFERICH:100:2:prpnet.primegrid.com:13000 NOTE: For those new to PRPNet, please see the Welcome to PRPNet post.
Stats can be viewed here: Wiefrich Prime Search
Additional Information
For more in depth information, please see the following:
 Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997), "A search for Wieferich and Wilson primes", Math. Comp. 66 (217):433–449.
 Dorais, F. G.; Klyve, D. (2011), "A Wieferich Prime Search Up to 6.7×10^15", Journal of Integer Sequences 14 (Article 11.9.2).
 Knauer, Joshua; Richstein, Jörg (2005), "The continuing search for Wieferich primes", Math. Comp. 74 (251):1559–1563.
 McIntosh, R. J. (2004), WallSunSun (Fibonacci Wieferich) Search Status email to Paul Zimmermann.
 

JimBVolunteer moderator Project developer Project scientist Send message
Joined: 4 Aug 11 Posts: 623 ID: 107307 Credit: 424,891,226 RAC: 2,373,897

Welcome to the WallSunSun Prime Search
A Wall–Sun–Sun (or Fibonacci–Wieferich) prime is a prime p > 5 in which p^2 divides the Fibonacci number , where the Legendre symbol is defined as
Although it has been conjectured that infinitely many exist, there are no known Wall–Sun–Sun primes. As of December 2011, if any exist, they must be > 9.7e14. We will begin our search at this limit.
The lack of success has lead to an interest in "Near" WallSun_Sun primes. They are defined as special instances (with small A) of F_(p(p/5)) (mod p^2). Here's a list of "Near" WallSun_Sun primes (coming soon) with p > 200.
Search History
Search limit Author Year
1e9 Williams 1982
2^32 Montgomery 1991
1e14 Knauer and McIntosh 2003
2e14 McIntosh and Roettger 2005
9.7e14 Dorais and Klyve 2011
10e14 PrimeGrid 20111228
15e14 PrimeGrid 20120110
20e14 PrimeGrid 20120122
25e14 PrimeGrid 20120302
60e14 PrimeGrid 20120729
28e15 PrimeGrid 20140331
They are named after Donald Dines Wall and twin brothers ZhiHong Sun and ZhiWei Sun. Drawing on Wall's work, in 1992 the brothers proved that if the first case of Fermat's last theorem was false for a certain prime p, then that p would have to be a Wall–Sun–Sun prime.
Classical Definition of nearness
A prime p satisfying the congruence F_(p(p/5)) ≡ Ap (mod p^2) with small A is commonly called a nearWallSunSun prime. Therefore, we are going to classify finds as follows:
WallSunSun prime: A = 0
A <= 10
A <= 100
A <= 1000
For a list of nearWallSunSun primes meeting the above criteria see here:
WallSunSun  sortable
WallSunSun  pdf How to Participate
To participate in the search, add the following server/port to your prpclient.ini file (adjust project share accordingly):
server=WALLSUNSUN:100:2:prpnet.primegrid.com:13001 NOTE: For those new to PRPNet, please see the Welcome to PRPNet post.
Stats can be viewed here: Wiefrich Prime Search
Additional Information
For more in depth information, please see the following:
 McIntosh, R. J. (2004), WallSunSun (Fibonacci Wieferich) Search Status email to Paul Zimmermann.
 Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997), "A search for Wieferich and Wilson primes", Math. Comp. 66 (217):433–449.
 McIntosh, R. J.; Roettger, E. L. (2007), "A search for FibonacciWieferich and Wolstenholme primes", Math. Comp. 76 (260):2087–2094.
 Dorais, F. G.; Klyve, D. (2011), "A Wieferich Prime Search Up to 6.7×10^15", Journal of Integer Sequences 14 (Article 11.9.2)..
 

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