Integer sequence prime

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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 is called an e-prime.

Other examples of integer sequence primes include:

  • Cullen prime – a prime that appears in the sequence of Cullen numbers a_{n}=n2^{n}+1\,.
  • Factorial prime – a prime that appears in either of the sequences a_{n}=n!-1 or b_{n}=n!+1\,.
  • Fermat prime – a prime that appears in the sequence of Fermat numbers a_{n}=2^{{2^{n}}}+1\,.
  • Fibonacci prime – a prime that appears in the sequence of Fibonacci numbers.
  • Lucas prime – a prime that appears in the Lucas numbers.
  • Mersenne prime – a prime that appears in the sequence of Mersenne numbers a_{n}=2^{n}-1\,.
  • Primorial prime – a prime that appears in either of the sequences a_{n}=n\#-1 or b_{n}=n\#+1\,.
  • Pythagorean prime – a prime that appears in the sequence a_{n}=4n+1\,.
  • Woodall prime – a prime that appears in the sequence of Woodall numbers a_{n}=n2^{n}-1\,.

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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