Fermat number: Difference between revisions
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{{one-parameter sequence}} | {{one-parameter sequence}} | ||
{{natural number property}} | |||
==Definition== | ==Definition== | ||
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If it is prime, it is termed a [[Fermat prime]]. | If it is prime, it is termed a [[Fermat prime]]. | ||
==Occurrence== | |||
===Initial values=== | |||
The initial values of <math>F_n, n = 0,1,2,3,4,\dots</math> are <section begin="list"/>[[3]], [[5]], [[17]], [[257]], [[65537]], 4294967297 [[Oeis:A000215|View list on OEIS]]<section end="list"/> | |||
==Facts== | |||
* [[Composite Fermat number implies Poulet number]]: This states that any composite Fermat number is a [[Poulet number]], i.e., a [[Fermat pseudoprime]] to base 2. | |||
* [[Prime divisor of Fermat number is congruent to one modulo large power of two]]: For a Fermat number <math>F_k = 2^{2^k} + 1</math>, any prime divisor is congruent to 1 mod <math>2^{k+1}</math>, and for <math>k \ge 2</math>, congruen to 1 mod <math>2^{k+2}</math>. | |||
* [[Quadratic nonresidue equals primitive root for Fermat prime]] | |||
==Relation with other properties== | ==Relation with other properties== | ||
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* [[Safe prime]] is a prime <math>p</math> such that <math>(p-1)/2</math> is also prime. | * [[Safe prime]] is a prime <math>p</math> such that <math>(p-1)/2</math> is also prime. | ||
* [[Mersenne number]] is a number of the form <math>2^n - 1</math>, and a [[Mersenne prime]] is a Mersenne number that is also prime. | * [[Mersenne number]] is a number of the form <math>2^n - 1</math>, and a [[Mersenne prime]] is a Mersenne number that is also prime. | ||
Latest revision as of 18:36, 3 July 2012
This article describes a sequence of natural numbers. The parameter for the sequence is a positive integer (or sometimes, nonnegative integer).
View other one-parameter sequences
This article defines a property that can be evaluated for a natural number, i.e., every natural number either satisfies the property or does not satisfy the property.
View a complete list of properties of natural numbers
Definition
Let be a nonnegative integer. The Fermat number, denoted , is defined as:
.
If it is prime, it is termed a Fermat prime.
Occurrence
Initial values
The initial values of are
3, 5, 17, 257, 65537, 4294967297 View list on OEIS
Facts
- Composite Fermat number implies Poulet number: This states that any composite Fermat number is a Poulet number, i.e., a Fermat pseudoprime to base 2.
- Prime divisor of Fermat number is congruent to one modulo large power of two: For a Fermat number , any prime divisor is congruent to 1 mod , and for , congruen to 1 mod .
- Quadratic nonresidue equals primitive root for Fermat prime
Relation with other properties
Weaker properties
- Safe prime is a prime such that is also prime.
- Mersenne number is a number of the form , and a Mersenne prime is a Mersenne number that is also prime.