Fermat prime

Fermat prime

      prime number (prime) of the form 22n + 1, for some positive integer n. For example, 223 + 1 = 28 + 1 = 257 is a Fermat prime. On the basis of his knowledge that numbers of this form are prime for values of n from 1 through 4, the French mathematician Pierre de Fermat (Fermat, Pierre de) (1601–65) conjectured that all numbers of this form are prime. However, the Swiss mathematician Leonhard Euler (Euler, Leonhard) (1707–83) showed that Fermat's conjecture is false for n = 5: 225 + 1 = 232 + 1 = 4,294,967,297, which is divisible by 641. In fact, it is known that numbers of this form are not prime for values of n from 5 through 30, placing doubt on the existence of any Fermat primes for values of n > 4.

* * *


Universalium. 2010.

Поможем написать реферат

Look at other dictionaries:

  • Prime number — Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is… …   Wikipedia

  • Fermat number — In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form:F {n} = 2^{2^{ overset{n} {} + 1where n is a nonnegative integer. The first nine Fermat numbers are OEIS|id=A000215:As of|2008 …   Wikipedia

  • Fermat, Pierre de — born Aug. 17, 1601, Beaumont de Lomagne, France died Jan. 12, 1665, Castres French mathematician. Of Basque origin, Fermat studied law at Toulouse and developed interests in foreign languages, Classical literature, ancient science, and… …   Universalium

  • Fermat'sche Primzahl — Eine Fermat Zahl, benannt nach dem französischen Mathematiker Pierre de Fermat, ist eine Zahl der Form wobei n eine natürliche Zahl ist. Die ersten Fermat Zahlen sind 3, 5, 17, 257, 65537, … (Folge A000215 in OEIS) Eine Fermat Zahl, die… …   Deutsch Wikipedia

  • Fermat-Zahl — Eine Fermat Zahl, benannt nach dem französischen Mathematiker Pierre de Fermat, ist eine Zahl der Form wobei n eine nichtnegative ganze Zahl ist. Die ersten Fermat Zahlen sind 3, 5, 17, 257, 65537, … (Folge A000215 in OEIS) Eine Fermat Zahl, die… …   Deutsch Wikipedia

  • prime — primeness, n. /pruym/, adj., n., v., primed, priming. adj. 1. of the first importance; demanding the fullest consideration: a prime requisite. 2. of the greatest relevance or significance: a prime example. 3. of the highest eminence or rank: the… …   Universalium

  • Fermat's Last Theorem — is the name of the statement in number theory that:: It is impossible to separate any power higher than the second into two like powers,or, more precisely:: If an integer n is greater than 2, then the equation a^n + b^n = c^n has no solutions in… …   Wikipedia

  • Fermat's little theorem — (not to be confused with Fermat s last theorem) states that if p is a prime number, then for any integer a , a^p a will be evenly divisible by p . This can be expressed in the notation of modular arithmetic as follows::a^p equiv a pmod{p},!A… …   Wikipedia

  • Fermat's factorization method — is based on the representation of an odd integer as the difference of two squares: :N = a^2 b^2. That difference is algebraically factorable as (a+b)(a b); if neither factor equals one, it is a proper factorization of N .Each odd number has such… …   Wikipedia

  • Fermat primality test — The Fermat primality test is a probabilistic test to determine if a number is probably prime. ConceptFermat s little theorem states that if p is prime and 1 le a < p, then :a^{p 1} equiv 1 pmod{p}If we want to test if p is prime, then we can pick …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”