Editing Prime number (section)

== Definition and examples ==
{{main|List of prime numbers}}
A [[natural number]] (1, 2, 3, 4, 5, 6, etc.) is called a ''prime number'' (or a ''prime'') if it is greater than 1 and cannot be written as the product of two smaller natural numbers. The numbers greater than 1 that are not prime are called [[composite number]]s.<ref>{{cite book|title=The Mathematical Olympiad Handbook: An Introduction to Problem Solving Based on the First 32 British Mathematical Olympiads 1965–1996|first=Anthony|last=Gardiner|author-link=Tony Gardiner|publisher=Oxford University Press|year=1997|isbn=978-0-19-850105-3|page=[https://archive.org/details/mathematicalolym1997gard/page/26 26]|url=https://archive.org/details/mathematicalolym1997gard|url-access=registration}}</ref> In other words, {{tmath|n}} is prime if {{tmath|n}} items cannot be divided up into smaller equal-size groups of more than one item,<ref>{{cite book|title=Dyslexia, Dyscalculia and Mathematics: A practical guide|first=Anne|last=Henderson|edition=2nd|publisher=Routledge|year=2014|isbn=978-1-136-63662-2|page=62|url=https://books.google.com/books?id=uy-yGVRUilMC&pg=PA62}}</ref> or if it is not possible to arrange {{tmath|n}} dots into a rectangular grid that is more than one dot wide and more than one dot high.<ref>{{cite book|title=The Giant Golden Book of Mathematics: Exploring the World of Numbers and Space|url=https://archive.org/details/giantgoldenbooko00adle|url-access=registration |first=Irving|last=Adler|author-link=Irving Adler|publisher=Golden Press|year=1960|page=[https://archive.org/details/giantgoldenbooko00adle/page/16 16]|oclc=6975809}}</ref> For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers,<ref>{{cite book | url = https://archive.org/details/barronsmathworkb00leff_0 | url-access = registration | page = [https://archive.org/details/barronsmathworkb00leff_0/page/360 360] | title = Math Workbook for the SAT I | first = Lawrence S. | last = Leff | publisher = Barron's Educational Series | year = 2000 | isbn = 978-0-7641-0768-9}}</ref> as there are no other numbers that divide them evenly (without a remainder). 1 is not prime, as it is specifically excluded in the definition. {{nowrap|1=4 = 2 × 2}} and {{nowrap|1=6 = 2 × 3}} are both composite.

[[File:Prime number Cuisenaire rods 7.png|thumb|upright=1.2|Demonstration, with [[Cuisenaire rods]], that 7 is prime, because none of 2, 3, 4, 5, or 6 divide it evenly|alt=refer to caption]]
The [[divisor]]s of a natural number {{tmath|n}} are the natural numbers that divide {{tmath|n}} evenly. Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This leads to an equivalent definition of prime numbers: they are the numbers with exactly two positive [[divisor]]s. Those two are 1 and the number itself. As 1 has only one divisor, itself, it is not prime by this definition.<ref>{{cite book | last1=Dudley | first1=Underwood | author-link=Underwood Dudley | title=Elementary number theory | publisher=W.H. Freeman and Co. | edition=2nd | isbn=978-0-7167-0076-0 | year=1978 | contribution=Section 2: Unique factorization | page=[https://archive.org/details/elementarynumber00dudl_0/page/10 10] | contribution-url=https://books.google.com/books?id=tr7SzBTsk1UC&pg=PA10 | url=https://archive.org/details/elementarynumber00dudl_0/page/10 }}</ref> Yet another way to express the same thing is that a number {{tmath|n}} is prime if it is greater than one and if none of the numbers <math>2, 3, \dots, n-1</math> divides {{tmath|n}} evenly.<ref>{{cite book | last = Sierpiński | first = Wacław | author-link = Wacław Sierpiński | title = Elementary Theory of Numbers | edition = 2nd | volume = 31 | series = North-Holland Mathematical Library | publisher = Elsevier | year = 1988 | isbn = 978-0-08-096019-7 | page = 113 | url = https://books.google.com/books?id=ktCZ2MvgN3MC&pg=PA113 }}</ref>

The first 25 prime numbers (all the prime numbers less than 100) are:<!--Do not add 1 to this list. Its exclusion from the list is addressed in the "History of prime numbers" section below.--><ref name=ziegler>{{cite journal | last = Ziegler | first = Günter M. | author-link = Günter M. Ziegler | issue = 4 | journal = [[Notices of the American Mathematical Society]] | mr = 2039814 | pages = 414–416 | title = The great prime number record races | volume = 51 | year = 2004}}</ref>
: [[2]], [[3]], [[5]], [[7]], [[11 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[29 (number)|29]], [[31 (number)|31]], [[37 (number)|37]], [[41 (number)|41]], [[43 (number)|43]], [[47 (number)|47]], [[53 (number)|53]], [[59 (number)|59]], [[61 (number)|61]], [[67 (number)|67]], [[71 (number)|71]], [[73 (number)|73]], [[79 (number)|79]], [[83 (number)|83]], [[89 (number)|89]], [[97 (number)|97]] {{OEIS|id=A000040}}.

No [[even number]] {{tmath|n}} greater than 2 is prime because any such number can be expressed as the product {{tmath| 2\times n/2 }}. Therefore, every prime number other than 2 is an [[odd number]], and is called an ''odd prime''.<ref>{{cite book|title=Numbers and Geometry|series=Undergraduate Texts in Mathematics|first=John|last=Stillwell|author-link=John Stillwell|publisher=Springer|year=1997|isbn=978-0-387-98289-2|page=9|url=https://books.google.com/books?id=4elkHwVS0eUC&pg=PA9}}</ref> Similarly, when written in the usual [[decimal]] system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5.<ref>{{cite book | last = Sierpiński | first = Wacław | author-link = Wacław Sierpiński | location = New York | mr = 0170843 | page = [https://archive.org/details/selectionproblem00sier/page/n37 40] | publisher = Macmillan | title = A Selection of Problems in the Theory of Numbers | url = https://archive.org/details/selectionproblem00sier | url-access = limited | year = 1964}}</ref>

The [[Set (mathematics)|set]] of all primes is sometimes denoted by <math>\mathbf{P}</math> (a [[boldface]] capital P)<ref>{{cite book|title=Elementary Methods in Number Theory|volume=195|series=Graduate Texts in Mathematics|contribution=Notations and Conventions|contribution-url=https://books.google.com/books?id=sE7lBwAAQBAJ&pg=PP10|first=Melvyn B.|last=Nathanson|author-link=Melvyn B. Nathanson|publisher=Springer|year=2000|isbn=978-0-387-22738-2|mr=1732941}}</ref> or by <math>\mathbb{P}</math> (a [[blackboard bold]] capital P).<ref>{{cite book|title=The Mathematics of Infinity: A Guide to Great Ideas|volume=111|series=Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts|first=Theodore G.|last=Faticoni|edition=2nd|publisher=John Wiley & Sons|year=2012|isbn=978-1-118-24382-4|page=44|url=https://books.google.com/books?id=I433i_ZGxRsC&pg=PA44}}</ref>