Integer sequence prime
Encyclopedia
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 number
s that are prime.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an integer sequence prime is a prime number
Prime number
A prime number 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 called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
found as a member of an integer sequence
Integer sequence
In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms...
. For example, the 8th Delannoy number
Delannoy number
In mathematics, a Delannoy number D describes the number of paths from the southwest corner of a rectangular grid to the northeast corner , using only single steps north, northeast, or east....
, 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
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
and considering prefixes of its decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
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
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
is called an e-prime.
Other examples of integer sequence primes include:
- Cullen prime – a prime that appears in the sequence of Cullen numbers
- Factorial primeFactorial primeA factorial prime is a prime number that is one less or one more than a factorial . The first few factorial primes are:n! − 1 is prime for :n! + 1 is prime for :...
– a prime that appears in either of the sequences
or 
- Fermat prime – a prime that appears in the sequence of Fermat numbers
- Fibonacci primeFibonacci primeA Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime.The first Fibonacci primes are :-Known Fibonacci primes:It is not known if there are infinitely many Fibonacci primes...
– a prime that appears in the sequence of Fibonacci numberFibonacci numberIn mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
s. - Lucas prime – a prime that appears in the Lucas numberLucas numberThe Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas , who studied both that sequence and the closely related Fibonacci numbers...
s. - Mersenne primeMersenne primeIn mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
– a prime that appears in the sequence of Mersenne numbers
- Primorial primePrimorial primeIn mathematics, primorial primes are prime numbers of the form pn# ± 1, where:The first few primorial primes are, the largest known primorial prime is 843301# - 1 with 365,851 digits, found in 2010 by the PrimeGrid project....
– a prime that appears in either of the sequences
or 
- Pythagorean primePythagorean primeA Pythagorean prime is prime number of the form 4n + 1. These are exactly the primes that can be the hypotenuse of a Pythagorean triangle.The first few Pythagorean primes are...
– a prime that appears in the sequence
- Woodall prime – a prime that appears in the sequence of Woodall numbers
The On-Line Encyclopedia of Integer Sequences
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...
includes many sequences corresponding to the prime subsequences of well-known sequences, for example A001605 for Fibonacci number
Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
s that are prime.