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prime number

A prime number is a whole number greater than 1, whose only two whole-number factors are 1 and itself.  The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.  

As we proceed in the set of natural numbers N = {1, 2, 3, ...}, the primes become less and less frequent in general.  However, there is no largest prime number.  For every prime number p, there exists a prime number p' such that p' is greater than p.  This was demonstrated in ancient times by the Greek mathematician Euclid.

Suppose n is a whole number, and we want to test it to see if it is prime.   First, we take the square root (or the 1/2 power) of n; then we round this number up to the next highest whole number.  Call the result m.  We must find all of the following quotients:

qm = n / m
q(m-1) = n / (m-1)
q(m-2) = n / (m-2)
q(m-3) = n / (m-3)
. . .
q3 = n / 3
q2 = n / 2

The number n is prime if and only if none of the q's, as derived above, are whole numbers.

A computer can be used to test extremely large numbers to see if they are prime.  But, because there is no limit to how large a natural number can be, there is always a point where testing in this manner becomes too great a task even for the most powerful supercomputers.  Various algorithms have been formulated in an attempt to generate ever-larger prime numbers.  These schemes all have limitations.

A Mersenne prime must be reducible to the form 2 n - 1, where n is a prime number. The first few known values of n that produce Mersenne primes are where n = 2, n = 3, n = 5, n = 7, n = 13, n = 17, n = 19, n = 31, n = 61, and n = 89.

A Fermat prime is a Fermat number that is also a prime number . A Fermat number n is of the form 2 m + 1, where m is the n th power of 2 (that is, m = 2 n , where n is an integer). 

This was last updated in June 2015

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