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[[File:Eratosthenes.jpg|thumb|[[Eratosthenes]].]]A '''prime number''' is a [[natural number]] greater than 1 that is [[divisible ]] by only 1 and itself. The only even Described another way, a prime number is has only two [[factor]]s, 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13 . There is no upper limit to the quantity ... The opposite of primes, but there is no known formula for deriving the nth a primenumber is a highly [[composite number]]. <br><br>[[Leonhard Euler]] once commentedwrote: <blockquote>

There are infinitely many primes. The number of primes smaller than a given number <math>N</math> is roughly <math>\frac{N}{\ln(N)}</math>, where <math>\ln(N)</math> is the only [[natural logarithmeven]] (base ''prime number is 2. The [[eBible]]'') of <math>N</math>includes many references to prime numbers.<ref>http://home.att.net/~numericana/answer/primes.htm</ref> This is a formulation of a more general statement known as the [[prime number theorem]].

==The Prime NumbersSieve of Eratosthenes==The smallest {|class="wikitable" align="right"|{{User:ClementB/SieveOfEratosthenes}}|-|Here, '''n''' equals 144|}To construct a table of all prime numbers are 2less than ''n'', 3, 5, 7, 11, 13.you can use a method called the [[Sieve of Eratosthenes]]. Simply write down an ordered list of all counting numbers from 1 to ''n''. Beginning with 2*2 = 4, cross out every second number:

:2 3 <s>4</s> 5 <s>6</s> 7 <s>8</s> 9 <s>10</s> 11 ... then beginning with 3*3 = 9 cross out every third number: :2 3 <s>4</s> 5 <s>6</s> 7 <s>8 9 10</s> 11 ... Beginning with 5*5 = 25, cross out every fifth number, then repeat for the primes 7, 11, 13, etc. until <math>\sqrt{n}</math>. The number of primes smaller than a given number <math>N</math> is roughly <math>\frac{N}{\ln(N)}</math>, where <math>\ln(N)</math> is the [[natural logarithm]] (base ''[[e]]'') of <math>N</math>.<ref>http://home.att.net/~numericana/answer/primes.htm</ref> This is a formulation of a more general statement known as the [[Prime Number Theorem]]. ==How many prime numbers=={{main|Prime Number Theorem}} It is easy to prove that there are an infinite number of primes using Euclid's second theorem: Imagine there is a finite set consisting of all the primes. Multiply them all together, add 1, and call this N. N would not be divisible by any number in the set--there would always be a remainder of 1. Because all non-prime numbers can be decomposed into a product of underlying primes(by the [[Fundamental Theorem of Arithmetic]]), N must be divisible by some prime not in the set (possibly itself), thus contradicting the assumption that the set contained all of the primes.

To construct A '''Mersenne prime''' is a table of all prime numbers less than number ''M'' that is of the form ''M = 2ⁿ − 1'', you would use a method called the where ''n'' is some [[Sieve of Eratosthenesprime number]]<ref>[http://mathworld.wolfram.com/MersennePrime.html Mersenne Prime] from Wolfram</ref>. Simply write down an ordered list As of all counting numbers greater than 12015, only 48 Mersenne Primes have been found. <ref>http://primes.utm.edu/largest.html#largest</ref> Beginning with The largest known prime, ''2*2 = 4^57885161-1'', cross out every second number:is a Mersenne Prime. The concept of a Mersenne Prime was discovered by the French theologian, philosopher and mathematician [[Marin Mersenne]].

then beginning with 3*3 = 9 cross out According to the [[Fundamental Theorem of Arithmetic]], proven by [[Carl Friedrich Gauss]], every third number: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 <s>4</s> 5 <s>6</s> 7 <s>8 9 10</s> * 11 ..There is no other product of primes that equals 132.

Beginning with 5*5 = 25, cross out every fifth number, then repeat Finding the prime factors for large numbers (having hundreds of digits) can take considerable time (millions of years) even with the primes 7, 11, 13, etc. until <math>\sqrt{n}</math>most advanced computers.

Mathematicians often prefer Beyond the following definitionintegers, whichmathematicians 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 integers, is equivalent to the one stated abovea "prime" element: <blockquote>''A prime number 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>

The advantage of For the integers, this formulation definition is that it can be generalized on other structures which allow for addition and multiplication, iequivalent to the one stated above.e., [[ring]]sHowever in general the two definitions are not equivalent. The earlier definition involves is still important in rings, and is given the notion of name [[irreducibility]], since it literally states that the element cannot be broken into smaller pieces.

==Unique Factorization==According The preference for this alternate definition of "prime" is that this definition gives you unique [[factorization]] theorems analogous to the [[Fundamental Theorem of Arithmetic]]. For more information, proven by [[Carl Friedrich Gauss]]consult an abstract algebra book<ref>'''Abstract Algebra: An Introduction''', every positive integer has a unique factorization into prime numbersThomas Hungerford. 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 = Brooks Cole; 2 * 2 * 3 * 11. There is no other product of primes that equals 132. Finding the prime factors for large numbers can take considerable time edition (millions of years1996) even with the most advanced computers. This is referred to as the prime factorization problem, and it is believed to be [[NP-complete]].</ref>

To determine whether a number is prime, you can use the trial division method, provided the number you are testing is not too large. Let ''N'' be the number being tested and ''s'' be its square root. Divide ''N'' by each number in the prime table, beginning with 2. If there is no remainder, stop--''N '' is composite. Otherwise, test again with the next largest prime in the table. If the next largest prime is greater than ''s'', stop--''N'' is prime.

==The Largest Known Open Problems about PrimeNumbers==

==Mersenne Prime=Twin primes=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]].={{main|twin primes}}

==Riemann hypothesis==Most mathematicians believe a proof of the [[Riemann hypothesisTwin 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 best hope for discovering some pattern in closest two primes can be to each other, with the distribution exception of prime numbers2 and 3. ''The Clay Mathematics Institute'' has offered a one million dollar prize for a proof of the Riemann hypothesisfirst few twin primes are 3 and 5, 5 and 7, 11 and 13, 17 and 19, and 29 and 31.<ref>http://www.claymath.org/millennium/Riemann_Hypothesis/</ref>

==Goldbach's 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 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 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.

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== * [[Composite number]]* [[Fibonacci sequence]]* [[Perfect number]] ==External links==*[http://www.mersenne.org/ ''Great Internet Mersenne Prime Search'' project website]*[http://primes.utm.edu/ Prime Number resource Archive]*[http://educ.queensu.ca/%7Efmc/december2003/Sieve.html Sieve of Eratosthenes]

==External Links==:[http://www.mersenne.org/ ''Great Internet Mersenne Prime Search'' project website]:[httpCategory://primes.utm.edu/ Prime Number resource ArchieveTheory]][[Category:MathematicsFeatured articles]]