Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Prime number
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
=== 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>
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)