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Fermat Numbers have the form: where n is a nonnegative integer. They are named after Pierre de Fermat (1601-1665), a French Lawyer who first studied them. Fermat is casually referred to as a math hobbiest and is often called the Prince of Amateurs. However, his contributions to mathematics greatly outweigh those references. He was clearly a gifted mathematician who is given credit for early developments that led to modern calculus and is often called the founder of modern number theory. He is best known for the conjecture, now proved, known as Fermat's last theorem which states that: x^n + y^n = z^n has no non-zero integer solutions for x, y and z when n is greater than 2. It was proved in 1993 by Andrew J. Wiles
However, the topic at hand is about Fermat numbers: Fermat discovered that the first 5 numbers of this form are prime F(0)=3, F(1)=5, F(2)=17, F(3)=257, and F(4)=65537. These are called Fermat Primes. Although he had no proof, Fermat conjectured that all numbers of this form were prime. However, in 1732 Leonhard Euler showed that F(5) had a factor: 641 and was therefore not prime; F(5)=2^2^5 + 1 = 2^32 + 1 = 4,294,967,297 = 641 × 6,700,417. From this date, the search for these rare numbers (Fermat Number divisors) began. Since then, only 270 divisors of Fermat Numbers have been discovered. Still today, only those first 5 Fermat Numbers are known to be prime.
Euler proved that every factor of a Fermat Number F(n) must have the form k*2^(n+1) + 1 for n greater than 2. For n=5, this means that the only possible factors are of the form k*2^(5+1) +1 = k*2^6 +1 = k*64 +1. Euler found the factor 641 = 10×64 + 1. The simple form of this is k*2^n +1. Thus, when a prime has the form k*2^n +1, it has the potential to be a Fermat Number divisor.
Currently, these primes are being found in PrimeGrid's Proth Prime Search. After each prime is found, external (to BOINC) testing is done using OpenPFGW to test for Fermat Number divisibility.
"It appears that the probability of each prime of the form k.2^n+1 dividing a Fermat number is 1/k." (Harvey Dubner & Wilfrid Keller, "Factors of generalized Fermat numbers", Mathematics of Computation, Vol. 64, Number 209, January 1995, pp. 397-405).
Numbers of the form a^(2^m) + b^(2^m), where a > 1 are called generalized Fermat Numbers. Divisors to these numbers are much more numerous. Divisibility of generalized Fermat Numbers are tested at the same time as Fermat Number divisibility.
Wilfrid Keller keeps a current and detailed account of all known divisors of Fermat Numbers and generalized Fermat Numbers. They can be found here:
The discovery was made by Eric Ueda of the United States using an Intel C2Q Q6600 @ 2.40 GHz with 1 GB RAM. This computer took almost 4 minutes 43 seconds to test. Eric is a member of TeAm AnandTech.
The prime was verified on 27 Dec 2008 23:56:34 UTC, by Jeff Smith of Canada using an Intel C2Q Q9450 @ 2.66 GHz with 2 GB RAM. This computer took almost 4 minutes 20 seconds to test. Beta-guy is a member of team Canada.
The credits for the discovery are as follows:
1. Eric Ueda (United States), discoverer
2. PrimeGrid, et al.
3. Srsieve, sieving program developed by Geoff Reynolds
4. LLR, primality program developed by Jean Penn�
OpenPFGW, a primality program developed by Chris Nash & Jim Fougeron was used to check for Fermat Number divisibility using the following settings: -gxo -a1 651*2^476632+1. OpenPFGW's bio page at the Prime Pages can be found here: OpenPFGW. Also, for more information about Fermat and Generalized Fermat Number divisors, please see Wilfrid Keller's sites:
The discovery was made by Senji Yamashita (s-yama) of Japan using an Intel C2Q Q9450 @ 2.66GHz with 2 GB RAM. This computer took about 8 minutes to test. Senji is a member of team Tamagawa Data Center.
The prime was verified on 6 Mar 2009 7:42:18 UTC, by Iain Boutcher (Dorfl) of the United Kingdom using an Intel C2Q Q9550 @ 2.83GHz with 4 GB RAM. This computer took about 7 minutes 28 seconds to test.
The credits for the discovery are as follows:
1. Senji Yamashita (Japan), discoverer
2. PrimeGrid, et al.
3. Srsieve, sieving program developed by Geoff Reynolds
4. LLR, primality program developed by Jean Penn�
OpenPFGW, a primality program developed by Chris Nash & Jim Fougeron was used to check for Fermat Number divisibility using the following settings: -gxo -a1 519*2^567235+1. OpenPFGW's bio page at the Prime Pages can be found here: OpenPFGW. Also, for more information about Fermat and Generalized Fermat Number divisors, please see Wilfrid Keller's sites:
The discovery was made by Eric Embling (Eric E) of the United States using an Intel C2D E6750 @ 2.66GHz with 4 GB RAM. This computer took about 8 minutes 31 seconds to test. Eric is a member of team [H]ard|OCP.
The prime was verified on 1 Apr 2009 10:47:46 UTC, by (densibou@newsVIP) of Japan using an Intel C2D CPU E8500 @ 3.16GHz with 3 GB RAM. This computer took about 7 minutes 48 seconds to test. densibou@newsVIP is a member of team Team 2ch.
The credits for the discovery are as follows:
1. Eric Embling (United States), discoverer
2. PrimeGrid, et al.
3. Srsieve, sieving program developed by Geoff Reynolds
4. LLR, primality program developed by Jean Penn�
OpenPFGW, a primality program developed by Chris Nash & Jim Fougeron was used to check for Fermat Number divisibility using the following settings: -gxo -a1 659*2^617815+1. OpenPFGW's bio page at the Prime Pages can be found here: OpenPFGW. Also, for more information about Fermat and Generalized Fermat Number divisors, please see Wilfrid Keller's sites:
where n is a nonnegative integer. They are named after Pierre de Fermat (1601-1665), a French Lawyer who first studied them. Fermat is casually referred to as a math hobbiest and is often called the Prince of Amateurs. However, his contributions to mathematics greatly outweigh those references. He was clearly a gifted mathematician who is given credit for early developments that led to modern calculus and is often called the founder of modern number theory. He is best known for the conjecture, now proved, known as Fermat's last theorem which states that: x^n + y^n = z^n has no non-zero integer solutions for x, y and z when n is greater than 2. It was proved in 1993 by Andrew J. Wiles