Editing Prime number (section)

=== Constructible polygons and polygon partitions ===
[[File:Pentagon construct.gif|thumb|Construction of a regular pentagon using straightedge and compass. This is only possible because 5 is a [[Fermat prime]].|alt=Construction of a regular pentagon using straightedge and compass]]
[[Fermat prime]]s are primes of the form
: <math>F_k = 2^{2^k}+1,</math>
with {{tmath|k}} a [[nonnegative integer]].<ref>{{harvtxt|Boklan|Conway|2017}} also include {{tmath|1= 2^0+1=2 }}, which is not of this form.</ref> They are named after [[Pierre de Fermat]], who conjectured that all such numbers are prime. The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime,<ref name="kls">{{cite book | last1 = Křížek | first1 = Michal | last2 = Luca | first2 = Florian | last3 = Somer | first3 = Lawrence | doi = 10.1007/978-0-387-21850-2 | isbn =978-0-387-95332-8 | location = New York | mr = 1866957 | pages = 1–2 | publisher = Springer-Verlag | series = CMS Books in Mathematics | title = 17 Lectures on Fermat Numbers: From Number Theory to Geometry | url = https://books.google.com/books?id=hgfSBwAAQBAJ&pg=PA1 | volume = 9 | year = 2001}}</ref> but <math>F_5</math> is composite and so are all other Fermat numbers that have been verified as of 2017.<ref>{{cite journal | last1 = Boklan | first1 = Kent D. | last2 = Conway | first2 = John H. | author2-link = John Horton Conway | arxiv = 1605.01371 | date = January 2017 | doi = 10.1007/s00283-016-9644-3 | issue = 1 | journal = [[The Mathematical Intelligencer]] | pages = 3–5 | title = Expect at most one billionth of a new Ferma''t'' prime! | volume = 39 | s2cid = 119165671 }}</ref> A [[regular polygon|regular {{tmath|n}}-gon]] is [[constructible polygon|constructible using straightedge and compass]] if and only if the odd prime factors of {{tmath|n}} (if any) are distinct Fermat primes.<ref name="kls"/> Likewise, a regular {{tmath|n}}-gon may be constructed using straightedge, compass, and an [[Angle trisection|angle trisector]] if and only if the prime factors of [[regular polygon|{{tmath|n}}]] are any number of copies of 2 or 3 together with a (possibly empty) set of distinct [[Pierpont prime]]s, primes of the form {{tmath|2^a3^b+1}}.<ref>{{cite journal | last = Gleason | first = Andrew M. | author-link = Andrew M. Gleason | doi = 10.2307/2323624 | issue = 3 | journal = [[American Mathematical Monthly]] | mr = 935432 | pages = 185–194 | title = Angle trisection, the heptagon, and the triskaidecagon | volume = 95 | year = 1988| jstor = 2323624 }}</ref>

It is possible to partition any convex polygon into {{tmath|n}} smaller convex polygons of equal area and equal perimeter, when {{tmath|n}} is a [[prime power|power of a prime number]], but this is not known for other values of {{tmath|n}}.<ref>{{cite journal | last = Ziegler | first = Günter M. | author-link = Günter M. Ziegler | issue = 95 | journal = European Mathematical Society Newsletter | mr = 3330472 | pages = 25–31 | title = Cannons at sparrows | year = 2015}}</ref>