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<p><b>New page</b></p><div>[[File:Primes-vs-composites.svg|thumb|Primes-vs-composites]] [[File:Prime_number_Cuisenaire_rods_7.png|thumb|Prime number Cuisenaire rods 7|left]] [[File:Rhind_Mathematical_Papyrus.jpg|thumb|Rhind Mathematical Papyrus]] [[File:Prime-counting_relative_error.svg|thumb|Prime-counting relative error]] [[File:Ulam_2.png|thumb|Ulam 2]] [[File:Riemann_zeta_function_absolute_value.png|thumb|Riemann zeta function absolute value]] '''Prime number'''<br />
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A '''prime number''' is a [[natural number]] greater than 1 that has no positive [[divisor]]s other than 1 and itself. A prime number is a fundamental concept in [[number theory]] and has been studied for thousands of years. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.<br />
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== Properties ==<br />
Prime numbers have several important properties:<br />
* They are the building blocks of the [[integers]], as every integer greater than 1 can be uniquely factored into prime numbers, a concept known as the [[fundamental theorem of arithmetic]].<br />
* The number 2 is the only even prime number, as all other even numbers can be divided by 2.<br />
* There are infinitely many prime numbers, a fact that was first proven by the ancient Greek mathematician [[Euclid]].<br />
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== Distribution ==<br />
The distribution of prime numbers among the integers is a central topic in number theory. The [[prime number theorem]] describes the asymptotic distribution of prime numbers and states that the number of primes less than a given number \( n \) is approximately \( \frac{n}{\ln(n)} \).<br />
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== Prime Number Tests ==<br />
Several algorithms exist to test whether a number is prime:<br />
* [[Trial division]]: The simplest method, which involves dividing the number by all integers up to its square root.<br />
* [[Sieve of Eratosthenes]]: An ancient algorithm that efficiently finds all primes up to a given limit.<br />
* [[Miller-Rabin primality test]]: A probabilistic test that can quickly determine if a number is likely prime.<br />
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== Applications ==<br />
Prime numbers have numerous applications in modern [[cryptography]], particularly in [[public-key cryptography]] algorithms such as [[RSA (cryptosystem)|RSA]].<br />
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== Related Concepts ==<br />
* [[Composite number]]<br />
* [[Twin prime]]<br />
* [[Mersenne prime]]<br />
* [[Fermat prime]]<br />
* [[Goldbach's conjecture]]<br />
* [[Prime gap]]<br />
* [[Prime factorization]]<br />
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== See Also ==<br />
* [[List of prime numbers]]<br />
* [[Prime number theorem]]<br />
* [[Euclid's theorem]]<br />
* [[Riemann hypothesis]]<br />
* [[Sieve of Atkin]]<br />
* [[Sophie Germain prime]]<br />
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== References ==<br />
{{Reflist}}<br />
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== External Links ==<br />
{{Commons category|Prime numbers}}<br />
{{Wiktionary|prime number}}<br />
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[[Category:Number theory]]<br />
[[Category:Prime numbers]]<br />
[[Category:Mathematics]]<br />
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