section_path: Other applications/Constructible polygons and polygon partitions
section_name: Constructible polygons and polygon partitions
paragraph_text: Constructible polygons and polygon partitions
  
  Fermat primes are primes of the form
  
  with k {\displaystyle k} a nonnegative integer. 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, but F 5 {\displaystyle F_{5}} is composite and so are all other Fermat numbers that have been verified as of 2017. A regular n {\displaystyle n} -gon is constructible using straightedge and compass if and only if the odd prime factors of n {\displaystyle n} (if any) are distinct Fermat primes. Likewise, a regular n {\displaystyle n} -gon may be constructed using straightedge, compass, and an angle trisector if and only if the prime factors of n {\displaystyle n} are any number of copies of 2 or 3 together with a (possibly empty) set of distinct Pierpont primes, primes of the form 2 a 3 b + 1 {\displaystyle 2^{a}3^{b}+1}.
  
  It is possible to partition any convex polygon into n {\displaystyle n} smaller convex polygons of equal area and equal perimeter, when n {\displaystyle n} is a power of a prime number, but this is not known for other values of n {\displaystyle n}.