Editing Prime number (section)

{{Short description|Number divisible only by 1 or itself}}
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[[File:Primes-vs-composites.svg|thumb|[[Composite number]]s can be arranged into [[rectangle]]s but prime numbers cannot.|alt=Groups of two to twelve dots, showing that the composite numbers of dots (4, 6, 8, 9, 10, and 12) can be arranged into rectangles but prime numbers cannot]]

A '''prime number''' (or a '''prime''') is a [[natural number]] greater than 1 that is not a [[Product (mathematics)|product]] of two smaller natural numbers. A natural number greater than 1 that is not prime is called a [[composite number]]. For example, 5 is prime because the only ways of writing it as a product, {{nowrap|1 × 5}} or {{nowrap|5 × 1}}, involve 5 itself. However, 4 is composite because it is a product (2&nbsp;×&nbsp;2)<!-- {{nowrap}} avoided here so Navigation popups will display it --> in which both numbers are smaller than 4. Primes are central in [[number theory]] because of the [[fundamental theorem of arithmetic]]: every natural number greater than 1 is either a prime itself or can be [[factorization|factorized]] as a product of primes that is unique [[up to]] their order.

The property of being prime is called '''primality'''. A simple but slow [[primality test|method of checking the primality]] of a given number {{tmath|n}}, called [[trial division]], tests whether {{tmath|n}} is a multiple of any integer between 2 and {{tmath|\sqrt{n} }}. Faster algorithms include the [[Miller–Rabin primality test]], which is fast but has a small chance of error, and the [[AKS primality test]], which always produces the correct answer in [[polynomial time]] but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as [[Mersenne number]]s. {{As of|2024|10}} the [[largest known prime number]] is a Mersenne prime with 41,024,320 [[numerical digit|decimal digits]].<ref name="GIMPS-2024">{{cite web |title=GIMPS Discovers Largest Known Prime Number: 2<sup>136,279,841</sup> − 1 |url=https://www.mersenne.org/primes/?press=M136279841 |date=21 October 2024 |work=Mersenne Research, Inc. |access-date=21 October 2024}}</ref><ref>{{Cite journal |last=Sparkes |first=Matthew |date=November 2, 2024 |title=Amateur sleuth finds largest-known prime number |journal=New Scientist |page=19}}</ref>

There are [[Infinite set|infinitely many]] primes, as [[Euclid's theorem|demonstrated by Euclid]] around 300&nbsp;BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the [[prime number theorem]], proven at the end of the 19th century, which says roughly that the [[probability]] of a randomly chosen large number being prime is inversely [[proportionality (mathematics)|proportional]] to its number of digits, that is, to its [[logarithm]].

Several historical questions regarding prime numbers are still unsolved. These include [[Goldbach's conjecture]], that every even integer greater than 2 can be expressed as the sum of two primes, and the [[twin prime]] conjecture, that there are infinitely many pairs of primes that differ by two. Such questions spurred the development of various branches of number theory, focusing on [[analytic number theory|analytic]] or [[algebraic number theory|algebraic]] aspects of numbers. Primes are used in several routines in [[information technology]], such as [[public-key cryptography]], which relies on the difficulty of [[Integer factorization|factoring]] large numbers into their prime factors. In [[abstract algebra]], objects that behave in a generalized way like prime numbers include [[prime element]]s and [[prime ideal]]s.