Changes

Beginning with 5*5 = 25, cross out every fifth number, then repeat for the primes 7, 11, 13, etc. until <math>\sqrt{n}</math>.

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]].

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==

==How many prime numbers==

{{main|prime number theorem}}

{{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, 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.

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, 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.

==Unique Factorization==

==Unique Factorization==

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.

===Twin primes===

===Twin primes===

'''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 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.  

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 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).

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).

===Riemann hypothesis===

===Riemann hypothesis===

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>

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>

:[http://www.mersenne.org/ ''Great Internet Mersenne Prime Search'' project website]

:[http://www.mersenne.org/ ''Great Internet Mersenne Prime Search'' project website]

:[http://primes.utm.edu/ Prime Number resource Archieve]

:[http://primes.utm.edu/ Prime Number resource Archieve]

 

[[Category:Prime Numbers]]

[[Category:Number Theory]]