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In [[mathematics]], an '''integer sequence prime''' is a [[prime number]] found as a member of an [[integer sequence]]. For example, the 8th [[Delannoy number]], 265729, is prime. A challenge in empirical mathematics is to identify large prime values in rapidly growing sequences.
 
A common subclass of integer sequence primes are '''constant primes''', formed by taking a constant [[real number]] and considering prefixes of its [[decimal]] representation, omitting the decimal point. For example, the first 6 decimal digits of the constant ''[[π]]'', approximately 3.14159265, form the prime number 314159, which is therefore known as a '''pi-prime'''. Similarly, a constant prime based on ''[[e (mathematical constant)|e]]'' is called an '''e-prime'''.
 
Other examples of integer sequence primes include:
* [[Cullen prime]] &ndash; a prime that appears in the sequence of Cullen numbers <math>a_n=n2^n+1\, .</math>
* [[Factorial prime]] &ndash; a prime that appears in either of the sequences <math>a_n=n!-1</math> or <math>b_n=n!+1\, .</math>
* [[Fermat prime]] &ndash; a prime that appears in the sequence of Fermat numbers <math>a_n=2^{2^n}+1\, .</math>
* [[Fibonacci prime]] &ndash; a prime that appears in the sequence of [[Fibonacci number]]s.
* [[Lucas prime]] &ndash; a prime that appears in the [[Lucas number]]s.
* [[Mersenne prime]] &ndash; a prime that appears in the sequence of Mersenne numbers <math>a_n=2^n-1\, .</math>
* [[Primorial prime]] &ndash; a prime that appears in either of the sequences <math>a_n=n\#-1</math> or <math>b_n=n\#+1\, .</math>
* [[Pythagorean prime]] &ndash; a prime that appears in the sequence <math>a_n=4n+1\, .</math>
* [[Woodall prime]] &ndash; a prime that appears in the sequence of Woodall numbers <math>a_n=n2^n-1\, .</math>
 
The [[On-Line Encyclopedia of Integer Sequences]] includes many sequences corresponding to the prime subsequences of well-known sequences, for example [[OEIS:A001605|A001605]] for [[Fibonacci number]]s that are prime.
 
== References ==
 
* {{MathWorld | urlname=IntegerSequencePrimes | title=Integer Sequence Primes}}
* {{MathWorld | urlname=ConstantPrimes | title=Constant Primes}}
* {{MathWorld | urlname=Pi-Prime | title=Pi-Prime}}
* {{MathWorld | urlname=e-Prime | title=e-Prime}}
 
[[Category:Classes of prime numbers]]
 
 
{{Numtheory-stub}}

Revision as of 16:24, 9 January 2014

In mathematics, an integer sequence prime is a prime number found as a member of an integer sequence. For example, the 8th Delannoy number, 265729, is prime. A challenge in empirical mathematics is to identify large prime values in rapidly growing sequences.

A common subclass of integer sequence primes are constant primes, formed by taking a constant real number and considering prefixes of its decimal representation, omitting the decimal point. For example, the first 6 decimal digits of the constant π, approximately 3.14159265, form the prime number 314159, which is therefore known as a pi-prime. Similarly, a constant prime based on e is called an e-prime.

Other examples of integer sequence primes include:

The On-Line Encyclopedia of Integer Sequences includes many sequences corresponding to the prime subsequences of well-known sequences, for example A001605 for Fibonacci numbers that are prime.

References



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