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According to the [[Fundamental Theorem of Arithmetic]], proven by [[Carl Friedrich Gauss]], every positive integer has a unique [[factorization]] into prime numbers.

According to the [[Fundamental Theorem of Arithmetic]], proven by [[Carl Friedrich Gauss]], every positive integer has a unique [[factorization]] into prime numbers.

This means that every integer larger than 1, can be expressed as a product of one or more primes in only one way. For example, 132 = 2 * 2 * 3 * 11. There is no other product of primes that equals 132. This would not work if 1 were defined as a prime number.

This means that every integer larger than 1, can be expressed as a product of one or more primes in only one way. For example, 132 = 2 * 2 * 3 * 11. There is no other product of primes that equals 132.

Finding the prime factors for large numbers (having hundreds of digits) can take considerable time (millions of years) even with the most advanced computers.

Finding the prime factors for large numbers (having hundreds of digits) can take considerable time (millions of years) even with the most advanced computers.

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.

==Open Problems about Prime Numbers==

==Open Problems about Prime Numbers==

[[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.   

[[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.   

Twin primes become increasingly scarce 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 programs 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 debate is captured by the [[twin primes conjecture]]  

Twin primes become increasingly scarce 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 debate is captured by the [[twin primes conjecture]]  

Mathematicians have long been fascinated by twin primes and the biblically-approved relationships 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).

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).

===Goldbach's conjecture===

===Goldbach's conjecture===

The [[Riemann hypothesis]] is related to patterns 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>

The [[Riemann hypothesis]] is related to patterns 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==

The security of [[public-key cryptography]] schemes such as [[RSA]] depend on the difficulty of prime factorization, if a pattern in the distribution of prime numbers is discovered, then such [[cryptographic]] schemes could become vulnerable to attack. Cryptography is very important for keeping important data secure on the internet from atheist hackers.

The security of [[public-key cryptography]] schemes such as [[RSA]] depend on the difficulty of prime factorization, if a pattern in the distribution of prime numbers is discovered, then such [[cryptographic]] schemes could become vulnerable to attack.

==See also==

==See also==