PrimeGrid

Prime Numbers are of great interest to mathematicians for a variety of reasons. Primes also play a central role in the cryptographic systems which are used for computer security. Through the study of Prime Numbers it can be shown how much processing is required to crack an encryption code and thus to determine whether current security schemes are sufficiently secure.

PrimeGrid project URL; http://www.primegrid.com/

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PrimeGrid is currently running several sub-projects:

Twin Prime Search: searching for gigantic twin primes of the form k2n+1 and k2n−1

A gigantic prime is a prime number with at least 10,000 decimal digits.

Primegrid The term appeared in Journal of Recreational Mathematics in the article "Collecting gigantic and titanic primes" (1992) by Samuel Yates. Chris Caldwell, who continued Yates' collection in the prime pages, reports that he changed the requirement from Yates' original 5,000 digits to 10,000 digits, when he was asked to revise the article after the death of Yates. Few primes of that size were known then, but a modern PC can find many in a day.

The first discovered gigantic prime was the Mersenne prime 2⁴⁴⁴⁹⁷ − 1. It has 13,395 digits and was found in 1979 by Harry Nelson and David Slowinski.

The smallest gigantic prime is 10⁹⁹⁹⁹ + 33603. It was proved prime in 2003 by Jens Franke, Thorsten Kleinjung and Tobias Wirth with their own distributed ECPP program. It was the largest ECPP proof at the time.

A twin prime is a prime number that differs from another prime number by two. Except for the pair (2, 3), this is the smallest possible difference between two primes. Some examples of twin prime pairs are (5, 7), (11, 13), (17, 19), (29 ,31), (41, 43), and (821, 823). Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin.

The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture. A strong form of the twin prime conjecture, the Hardy-Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem.

Cullen-Woodall Search: searching for mega primes of forms n2n+1 and n2n−1

A megaprime is a prime number with at least one million decimal digits (whereas titanic prime is a prime number with at least 1000 digits, and gigantic prime has at least 10000 digits).

There are infinitely many primes and therefore infinitely many megaprimes. As of July 2008, seventeen megaprimes are known. The first known was the Mersenne prime 2^6972593−1 with 2,098,960 digits, discovered in 1999 by Nayan Hajratwala, a participant in the distributed computing project GIMPS.

321 Prime Search: searching for mega primes of the form 3*2n±1

Prime Sierpinski Project: helping Prime Sierpinski Project solve the Prime Sierpinski Problem

The Prime Sierpinski project (PSP) is a mathematical project involved in the search of large prime numbers. Prime numbers are numbers, which are divisible by 1 and by themselves and not by any other numbers. It has been proved that there are an infinite number of prime numbers but no one has been able to prove anything about the distribution of prime numbers in general. It is a mysterious frontier of mathematics.

Prime Sierpinski Project We look at a special class of prime numbers called proth numbers which have the general formula k*2^n+1. We further specialize our search by looking at numbers for which k is prime in k*2^n+1. Further more it has been proven that there exists an infinite number of prime k's such that k*2^n+1 can never be prime. These k's are called prime sierpinski numbers.

The smallest proven prime sierpinski number is 271129. We are looking at all prime k's below this number and trying to prove that they are not sierpinski numbers and thus studying the distribution of primes of the forum k*2^n+1. The easiest way to prove that a k is not a prime sierpinski number is to find a prime for that k.

There are currently 14 such candidates remaining for which we need to find a prime. We have already found 14 large primes, several of which made it into the top 100 largest known prime number list. Currently in this stage we are searching for primes up to n=50 million and once we reach there we plan to continue to higher values. There is a $100,000 prize given by the EFF corporation (www.eff.org) for finding a 10 million-digit prime. A 10 Million digit prime corresponds to n>34 million. We plan to find that 10 Million digit prime and win the prize. For this we need your help to find several primes and eliminate several more k's so that it becomes easier and easier to find a 10 Million digit prime.

The highest n limit of 50 Million was chosen because of efficiency reasons.
When a k is proved that it is not a sierpinski number the k is eliminated. This means that we no longer have to test that k for primality or find factors for this k. This makes the process faster (because of fewer numbers to test) and brings our goal of a 10 Million-digit prime closer to us. We saw no point in testing all the k's up to a higher n when most of them will produce a prime below n=50 million.