Fermat number

A Fermat number, named after the French mathematician Pierre de Fermat, a number of the form

Wherein a non-negative integer.

The first Fermat numbers are 3, 5, 17, 257, 65537, ....

Fermat primes

A Fermat number, which is the same prime number is called Fermat primes. Fermat showed that the first five Fermat numbers are prime numbers, and guessed in 1637 that this view for all Fermat numbers. This conjecture was disproved by Leonhard Euler in 1732 by finding a proper divisors of 641.

It is assumed now that there are no further apart from the first five Fermat primes. This assumption is based on the prime number theorem, according to which the number of primes that are not greater than, is approximately equal. The density of primes or " probability " that a prime number, is therefore approximately the sum of these expressions is a geometric series and for all not yet partly or fully factored Fermat numbers very small.

When a prime number, then it must be one or two. Otherwise, there would exist an odd divisor of the exponents and to having a decomposition into two factors, both of which are greater than 1, because and are positive.

Faktorisierungsstatus of Fermat numbers

The numbers are prime numbers up.

The fully factored Fermat numbers takes you to the following table.

From F5 to F7:

From F8:

Of the numbers F12 to F32 and some larger Fermat numbers is known that they are composed, mainly because one or more factors were found. From two Fermat numbers (F20 and F24) are known already not a factor, however, has shown in a different way that they are composed. For F14 was a factor 3 February 2010 published for F22 on 25 March 2010.

The smallest Fermat number, of which is not known whether it is prime or composite, is F33; F2478782 is the largest, of which one factor is known, namely the Pierpont prime He was discovered in 2003 by Cosgrave, Jobling, Woltman and Gallot. Overall, we know of 230 Fermat numbers that they are composed.

In order to prove by a Fermat number, that it is composed, one typically uses the Pépin - test and the Suyama test, both of which are especially tailored to this number and very fast.

Properties

• For a divisor p of a Fermat number Fn, n ≥ 5, p ≡ 1 (mod 2n 2) applies ( ie a factor of 641 F5: 641 = 27 * 5 1 = 128 * 5 1).

• Fermat numbers can be obtained by Fn = F0F1 ... Fn -1 2 calculated recursively.

• Two Fermat numbers are the same or prime, as follows from the last statement. It can be concluded that there are infinitely many primes (see also proof archive).

Geometric application of Fermat primes

Carl Friedrich Gauss showed that there is a correlation between the construction of regular polygons and Fermat primes: A regular polygon with sides can then and only then constructed with ruler and compass, if a power of 2 or the product of a power of 2 with different Fermat primes. In particular, Gauss showed that the constructability of the regular Siebzehnecks.

Generalized Fermat numbers

Instead of the base 2, you can also choose a different base. A number of the form with a natural number b is called generalized Fermat number. If such a number prime, then it is called generalized Fermat prime.

Example: b = 6, n = 2 gives the prime number

On November 19, 2011 at 14:03:58 UTC PrimeGrid PRPNet found the largest generalized Fermat primes currently with 2,558,647 digits. This therefore replaced the order 11:31:47 UTC by PrimeGrid using the search algorithm geneferCUDA found 29 October 2011, generalized Fermat prime biggest with 1,457,075 digits of the number one ranking.

Other well-known generalized Fermat primes are the prime published in March 2008 with 1,150,678 digits. and in 2003 discovered prime with 804 474 digits. The two prime numbers were discovered by the project Generalized Fermat Prime Search, generalized Fermat primes the great investigated.

Most of the above results could of course only be found with the help of computers.