Over the past nine months, six new projects (4 primality and 2 sieves) have been added to PrimeGrid. As mentioned in this post, "the primary focus was on simplicity...how easily could a new sub-project be implemented within PrimeGrid and BOINC."
We will soon be adding three new projects...all primality testing (LLR). Simplicity of implementation is still a driving factor right now. However, we may explore adding other primality programs in the future and add prime searches with increasing variety.
Sieving was conducted over the past several months and has been completed for the first project and ongoing for the other two projects.
Sophie Germain Prime Search
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. They are named after Marie-Sophie Germain, an extraordinary French mathematician.
We'll be searching the form k*2^n-1. If it is prime, then we'll check k*2^n+1, k*2^(n-1)-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^n-1, k*2^n+1, k*2^(n-1)-1, & k*2^(n+1)-1 do not have any small prime divisors.
As you can see, a twin prime is also possible from this search although we expect to find a Sophie Germain prime first. Here are some stats for the search:
k range: 1<k<41T
sieve depth: p=200T
candidates remaining: 34,190,344
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
Primes found in this search will enter the Top 5000 Primes database ranked about 600.
For more information about Sophie Germain primes, please visit these links:
For more infomation about Marie-Sophie Germain, please visit these links:
This will be a sister project to the already established 3*2^n-1 project. We hope to eventually have both projects at the same n value. We have reserved k=3 from the ProthSearch site. Our initial goal will be like 3*2^n-1, tested up to n=5M. However, sieving is currently being conducted beyond that.
Here are some stats for the search:
sieved n range: 1<n<5M
sieve depth: p=500T (ongoing)
3*2^n+1 will be a double check effort for even n up to ~1.8M and for odd n up to ~2.6M. Beyond that will be new primes, although there may be a small chance of a missed prime in the lower ranges.
+1 Prime Search
This search will be looking for primes in the form of k*2^n+1. With the condition 2^n > k, these are often called the Proth primes. We will be coordinating our effort through the ProthSearch site. This project will also have the added bonus of possibly finding Generalized Fermat Numbers (GFN) factors. Each k*2^n+1 prime found may be a GFN factor. As this requires PrimeFormGW (PFGW) (a primality-testing program), once PrimeGrid finds a prime, it will then be manually tested outside of BOINC for GFN divisibility.
Our initial goal will be to double check all previous work up to n=300K for k<1200 and to fill in any gaps that were missed. Primes found in this range will not make it into the Top 5000 Primes database (currently n>333333). However, the work is still important as it may lead to new GFN factors. Currently there are only about 250 such factors known.
Here are some stats for the search:
k range: 4<k<1200
n range: 1<n<5M
sieve depth: currently at p=10T (ongoing)
Once the initial goal is reached, we'll advance to n<400K and then n<500K. Afterwards, we'll turn our focus to smaller k values and higher n values. For example, k<32 complete to n=2M, k<64 complete to n=1M and so on. Primes found in these ranges will definitely make it into the Top 5000 Primes database.
For more information about "Proth" primes, please visit these links:
Other suggestions for future projects
Generalized Cullen/Woodall Search:
This is similar to our current Cullen/Woodall search except a base other than 2 will be selected. The form of these primes are as follows:
Generalized Cullen: n*b^n+1
Generalized Woodall: n*b^n-1
One base in particular, b=13, is interesting as no prime has yet to be found although it has been tested up to n=250K.
There are ongoing efforts here:
Steven Harvey's Generalized Woodall number Search
Günter Löh's Generalized Cullen Search for 3 <= b <= 100
Daniel Hermle's Generalized Cullen Search for 101 <= b <= 200
Again, similar to our current Cullen/Woodall search. The form of these primes are as follows:
HyperCullen: k^n*n^k+1, k>n
HyperWoodall: k^n*n^k-1, k>n
There is an ongoing effort here: Steven Harvey's Generalized Woodall number Search
Generalized Fermat Prime Search:
This searches for primes in the form b^2^n+1. A previous project has already completed a substantial amount of work. It can be found here: Generalized Fermat Prime Search. We may be able to double check all completed work and then help the previous project extend their search.
There is now an established effort for this search which can be found here: http://www.elmath.org/
There was an effort, but it is now on hiatus due to lack of interest. It can be found here: http://mersenneforum.org/forumdisplay.php?f=63
Riesel and Sierpinski conjectures:
There are two well known projects already established...Riesel Sieve and Seventeen or Bust. There is now an established effort for bases other than 2 which can be found here: http://mersenneforum.org/showthread.php?t=9738