Fermat prime - Revision history

Vipul: /* Facts */

2012-01-03T19:07:00Z

Facts

← Older revision		Revision as of 19:07, 3 January 2012
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	* [[Quadratic nonresidue equals primitive root for Fermat prime]]		* [[Quadratic nonresidue equals primitive root for Fermat prime]]
	* [[Fermat prime implies three is primitive root]] (we prove this by first showing that 3 is a quadratic nonresidue, then using the above).		* [[Fermat prime greater than three implies three is primitive root]] (we prove this by first showing that 3 is a quadratic nonresidue, then using the above).
	* [[Proth's theorem]]		* [[Proth's theorem]]
	* [[Pepin's primality test]]		* [[Pepin's primality test]]

Vipul: /* Facts */

2012-01-03T19:06:45Z

Facts

← Older revision		Revision as of 19:06, 3 January 2012
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	* [[Quadratic nonresidue equals primitive root for Fermat prime]]		* [[Quadratic nonresidue equals primitive root for Fermat prime]]
			* [[Fermat prime implies three is primitive root]] (we prove this by first showing that 3 is a quadratic nonresidue, then using the above).
	* [[Proth's theorem]]		* [[Proth's theorem]]
	* [[Pepin's primality test]]		* [[Pepin's primality test]]

Vipul at 22:31, 28 April 2009

2009-04-28T22:31:17Z

← Older revision		Revision as of 22:31, 28 April 2009
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	The only known Fermat primes are the Fermat primes for <math>k = 0,1,2,3,4</math>, namely, the primes <math>3,5,17,65537</math>. For all <math>k \ge 5</math>, either the [[Fermat prime]] <math>F_k</math> is known to be composite or its primality is open. The [[prime number theorem]] suggests that there are likely to be only finitely many Fermat primes, and it is conjectured that there are no Fermat primes other than the five known ones.		The only known Fermat primes are the Fermat primes for <math>k = 0,1,2,3,4</math>, namely, the primes <math>3,5,17,65537</math>. For all <math>k \ge 5</math>, either the [[Fermat prime]] <math>F_k</math> is known to be composite or its primality is open. The [[prime number theorem]] suggests that there are likely to be only finitely many Fermat primes, and it is conjectured that there are no Fermat primes other than the five known ones.

			{{oeis\|A019434}}

	==Relation with other properties==		==Relation with other properties==
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	==Testing==		==Testing==

	~~{{oeis\|A019434}}~~

	{{test\|Pepin's test}}		{{test\|Pepin's test}}

	Pepin's test is a deterministic primality test for Fermat numbers. It relies on [[Proth's theorem]], and the observation that <math>3</math> is a [[quadratic nonresidue]] for any Fermat prime greater than <math>3</math> itself.		Pepin's test is a deterministic primality test for Fermat numbers. It relies on [[Proth's theorem]], and the observation that <math>3</math> is a [[quadratic nonresidue]] for any Fermat prime greater than <math>3</math> itself.

Vipul at 23:13, 21 April 2009

2009-04-21T23:13:23Z

← Older revision		Revision as of 23:13, 21 April 2009
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	* [[Proth's theorem]]		* [[Proth's theorem]]
	* [[Pepin's primality test]]		* [[Pepin's primality test]]

			===Related facts about (not necessarily prime) Fermat numbers===

			* [[Goldbach's theorem]]: The Fermat numbers are pairwise relatively prime.
			* [[Prime divisor of Fermat number is congruent to one modulo large power of two]]

	==Occurrence==		==Occurrence==

Vipul: /* Facts */

2009-04-21T14:40:27Z

Facts

← Older revision		Revision as of 14:40, 21 April 2009
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	* [[Quadratic nonresidue equals primitive root for Fermat prime]]		* [[Quadratic nonresidue equals primitive root for Fermat prime]]
			* [[Proth's theorem]]
			* [[Pepin's primality test]]

	==Occurrence==		==Occurrence==

Vipul at 14:39, 21 April 2009

2009-04-21T14:39:36Z

← Older revision		Revision as of 14:39, 21 April 2009
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	It turns out that if <math>2^n + 1</math> is prime for <math>n</math> a natural number, then <math>n = 2^k</math> for some nonnegative integer <math>k</math>. Thus, a Fermat prime can also be defined as a prime of the form <math>2^n + 1</math> for some natural number <math>n</math>.		It turns out that if <math>2^n + 1</math> is prime for <math>n</math> a natural number, then <math>n = 2^k</math> for some nonnegative integer <math>k</math>. Thus, a Fermat prime can also be defined as a prime of the form <math>2^n + 1</math> for some natural number <math>n</math>.

			==Facts==

			* [[Quadratic nonresidue equals primitive root for Fermat prime]]

	==Occurrence==		==Occurrence==

Vipul at 19:56, 20 April 2009

2009-04-20T19:56:19Z

← Older revision		Revision as of 19:56, 20 April 2009
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	==Definition==		==Definition==

	A '''Fermat prime''' is a [[Fermat number]] that is also a [[prime number]]. In other words, it is a prime number of the form <math>F_k = 2^{2^k} + 1</math>, where <math>k</math> is a nonnegative integer.		A '''Fermat prime''' is a [[defining ingredient::Fermat number]] that is also a [[prime number]]. In other words, it is a prime number of the form <math>F_k = 2^{2^k} + 1</math>, where <math>k</math> is a nonnegative integer.

	It turns out that if <math>2^n + 1</math> is prime for <math>n</math> a natural number, then <math>n = 2^k</math> for some nonnegative integer <math>k</math>. Thus, a Fermat prime can also be defined as a prime of the form <math>2^n + 1</math> for some natural number <math>n</math>.		It turns out that if <math>2^n + 1</math> is prime for <math>n</math> a natural number, then <math>n = 2^k</math> for some nonnegative integer <math>k</math>. Thus, a Fermat prime can also be defined as a prime of the form <math>2^n + 1</math> for some natural number <math>n</math>.

			==Occurrence==

			===Finitude conjecture===

			{{further\|[[Finitude conjecture for Fermat primes]]}}

			The only known Fermat primes are the Fermat primes for <math>k = 0,1,2,3,4</math>, namely, the primes <math>3,5,17,65537</math>. For all <math>k \ge 5</math>, either the [[Fermat prime]] <math>F_k</math> is known to be composite or its primality is open. The [[prime number theorem]] suggests that there are likely to be only finitely many Fermat primes, and it is conjectured that there are no Fermat primes other than the five known ones.

	==Relation with other properties==		==Relation with other properties==
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	* [[Stronger than::Proth prime]]		* [[Stronger than::Proth prime]]

			===Complementary properties===

			* [[Composite Fermat number]]
	==Testing==		==Testing==

	{{oeis\|A019434}}		{{oeis\|A019434}}

			{{test\|Pepin's test}}

			Pepin's test is a deterministic primality test for Fermat numbers. It relies on [[Proth's theorem]], and the observation that <math>3</math> is a [[quadratic nonresidue]] for any Fermat prime greater than <math>3</math> itself.

Vipul at 16:04, 20 April 2009

2009-04-20T16:04:41Z

← Older revision		Revision as of 16:04, 20 April 2009
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	It turns out that if <math>2^n + 1</math> is prime for <math>n</math> a natural number, then <math>n = 2^k</math> for some nonnegative integer <math>k</math>. Thus, a Fermat prime can also be defined as a prime of the form <math>2^n + 1</math> for some natural number <math>n</math>.		It turns out that if <math>2^n + 1</math> is prime for <math>n</math> a natural number, then <math>n = 2^k</math> for some nonnegative integer <math>k</math>. Thus, a Fermat prime can also be defined as a prime of the form <math>2^n + 1</math> for some natural number <math>n</math>.

			==Relation with other properties==

			===Weaker properties===

			* [[Stronger than::Proth prime]]

			==Testing==

			{{oeis\|A019434}}

Vipul: Created page with '{{prime number property}} ==Definition== A '''Fermat prime''' is a Fermat number that is also a prime number. In other words, it is a prime number of the form $F_k...'$

2009-03-21T22:55:25Z

Created page with '{{prime number property}} ==Definition== A '''Fermat prime''' is a Fermat number that is also a prime number. In other words, it is a prime number of the form <math>F_k...'

New page

{{prime number property}}

==Definition==

A '''Fermat prime''' is a [[Fermat number]] that is also a [[prime number]]. In other words, it is a prime number of the form <math>F_k = 2^{2^k} + 1</math>, where <math>k</math> is a nonnegative integer.

It turns out that if <math>2^n + 1</math> is prime for <math>n</math> a natural number, then <math>n = 2^k</math> for some nonnegative integer <math>k</math>. Thus, a Fermat prime can also be defined as a prime of the form <math>2^n + 1</math> for some natural number <math>n</math>.

Fermat prime - Revision history

Vipul: /* Facts */

Vipul: /* Facts */

Vipul at 22:31, 28 April 2009

Vipul at 23:13, 21 April 2009

Vipul: /* Facts */

Vipul at 14:39, 21 April 2009

Vipul at 19:56, 20 April 2009

Vipul at 16:04, 20 April 2009

Vipul: Created page with '{{prime number property}} ==Definition== A '''Fermat prime''' is a Fermat number that is also a prime number. In other words, it is a prime number of the form F_k...'

Vipul: Created page with '{{prime number property}} ==Definition== A '''Fermat prime''' is a Fermat number that is also a prime number. In other words, it is a prime number of the form $F_k...'$