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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 2015, only 48 Mersenne Primes have been found.<ref>http://primes.utm.edu/largest.html#largest</ref>  The largest known prime, ''2^57885161-1'', 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 2015, only 48 Mersenne Primes have been found.<ref>http://primes.utm.edu/largest.html#largest</ref>  The largest known prime, ''2^57885161-1'', is a Mersenne Prime.  The concept of a Mersenne Prime was discovered by the French theologian, philosopher and mathematician [[Marin Mersenne]].

==Unique Factorization==

==Unique factorization==

{{main|Fundamental Theorem of Arithmetic}}

{{main|Fundamental Theorem of Arithmetic}}

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.

==Alternative Definition==

==Alternative definition==

Beyond the integers, mathematicians are interested in other structures with addition and multiplication; they are called [[Ring_(mathematics)|ring]]s. When working in rings, mathematicians use a different definition for a "prime" element: <blockquote>''A prime element is a non-unit (i.e., not 1 or -1 for the integers) which whenever it divides the product of two numbers will divide at least one of the factors.''</blockquote>

Beyond the integers, mathematicians are interested in other structures with addition and multiplication; they are called [[Ring_(mathematics)|ring]]s. When working in rings, mathematicians use a different definition for a "prime" element: <blockquote>''A prime element is a non-unit (i.e., not 1 or -1 for the integers) which whenever it divides the product of two numbers will divide at least one of the factors.''</blockquote>

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==