Changes

For larger values of ''N'', a method based on Fermat's theorem can be used, though it usually requires a computer.  Start with the number 1, and double it ''N-1'' times.  Divide this number by ''N'', keeping only the remainder.  If the remainder is greater than 1, ''N'' is composite.  Unfortunately, the converse is not true--if the remainder is 1, ''N'' is ''probably'' prime, but not necessarily.  Algorithms which improve on this method exist, and primality testing is today an active subject of mathematical research.

For larger values of ''N'', a method based on Fermat's theorem can be used, though it usually requires a computer.  Start with the number 1, and double it ''N-1'' times.  Divide this number by ''N'', keeping only the remainder.  If the remainder is greater than 1, ''N'' is composite.  Unfortunately, the converse is not true--if the remainder is 1, ''N'' is ''probably'' prime, but not necessarily.  Algorithms which improve on this method exist, and primality testing is today an active subject of mathematical research.

==The Largest Known Prime==

 

===The Largest Known Prime===

The largest known prime is ''2^32582657-1''; it was discovered by the '''Great Internet Mersenne Prime Search''' ('''GIMPS''')<ref>http://www.mersenne.org/</ref>, a distributed computing project launched by George Woltman in early 1996.<ref>http://primes.utm.edu/largest.html</ref>

The largest known prime is ''2^32582657-1''; it was discovered by the '''Great Internet Mersenne Prime Search''' ('''GIMPS''')<ref>http://www.mersenne.org/</ref>, a distributed computing project launched by George Woltman in early 1996.<ref>http://primes.utm.edu/largest.html</ref>

==Mersenne Prime==

===Mersenne Prime===

A '''Mersenne prime''' is a prime number ''M'' that is of the form ''M =  2ⁿ − 1'', where ''n'' is some [[prime number]]<ref>[http://mathworld.wolfram.com/MersennePrime.html Mersenne Prime] from Wolfram</ref>.  As of April, 2007, only 44 Mersenne Primes have been found.  The largest known prime ''2^32582657-1'', is a Mersenne Prime.  The concept of a Mersenne Prime was discovered by the French theologian, philosopher and mathematician [[Marin Mersenne]].

A '''Mersenne prime''' is a prime number ''M'' that is of the form ''M =  2ⁿ − 1'', where ''n'' is some [[prime number]]<ref>[http://mathworld.wolfram.com/MersennePrime.html Mersenne Prime] from Wolfram</ref>.  As of April, 2007, only 44 Mersenne Primes have been found.  The largest known prime ''2^32582657-1'', is a Mersenne Prime.  The concept of a Mersenne Prime was discovered by the French theologian, philosopher and mathematician [[Marin Mersenne]].

==Riemann hypothesis==

===Twin primes===

==Goldbach's conjecture==

'''Twin primes''' or '''prime twins''' are pairs of prime number which differ from each other by only 2.  Since all primes other than 2 are odd numbers, this is the closest two primes can be to each other, with the exception of 2 and 3.  The first few twin primes are 3 and 5, 5 and 7, 11 and 13, 17 and 19, and 29 and 31. 

Goldbach's conjecture asserts that every even [[integer]] greater than two is the sum of two [[prime]] numbers.  For example: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, and so on.

Twin primes become increasingly scarse among larger numbers.  For example, there are twenty-four sets of twin primes between 0 and 500, but only eleven sets between 500 and 1000.  Computer programmes have calculated some extremely large sets of twin primes.  While there are undoubtedly an unlimited number of primes, opinions differ among mathematicians whether there is also an infinite number of twin primes or whether there is an upper limit.  

This conjecture is one of the oldest unsettled statements in mathematics.  For centuries mathematicians have attempted to prove this statement, without success.  The conjecture is known to hold for numbers less than 3 <math>\cdot</math>10¹⁷.

Mathematicians have long been fascinated by twin primes and the relationship between them.  There seems to be no distinct pattern in their occurrence.  For example, there are no twin primes between 700 and 800, or between 900 and 1000, but there are five sets between 800 and 900 (809 and 811; 821 and 823; 827 and 829; 857 and 859; 881 and 883).

==Twin primes==

===Goldbach's conjecture===

'''Twin primes''' or '''prime twins''' are pairs of prime number which differ from each other by only 2.  Since all primes other than 2 are odd numbers, this is the closest two primes can be to each other, with the exception of 2 and 3.  The first few twin primes are 3 and 5, 5 and 7, 11 and 13, 17 and 19, and 29 and 31.

Goldbach's conjecture asserts that every even integer greater than two is the sum of two prime numbers.  For example: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, and so on.

Twin primes become increasingly scarse among larger numbers.  For example, there are twenty-four sets of twin primes between 0 and 500, but only eleven sets between 500 and 1000.  Computer programmes have calculated some extremely large sets of twin primes.  While there are undoubtedly an unlimited number of primes, opinions differ among mathematicians whether there is also an infinite number of twin primes or whether there is an upper limit.  

This conjecture is one of the oldest unsettled statements in mathematics.  For centuries mathematicians have attempted to prove this statement, without success.  The conjecture is known to hold for numbers less than 3 <math>\cdot</math>10¹⁷.

Mathematicians have long been fascinated by twin primes and the relationship between them.  There seems to be no distinct pattern in their occurrence.  For example, there are no twin primes between 700 and 800, or between 900 and 1000, but there are five sets between 800 and 900 (809 and 811; 821 and 823; 827 and 829; 857 and 859; 881 and 883).

Most mathematicians believe a proof of the [[Riemann hypothesis]] is the best hope for discovering some pattern in the distribution of prime numbers.  ''The Clay Mathematics Institute'' has offered a one million dollar prize for a proof of the Riemann hypothesis.<ref>http://www.claymath.org/millennium/Riemann_Hypothesis/</ref>

==Cryptography==

==Cryptography==