Gauss' proved that you can subdivide a circle into n
parts using a ruler (an unmarked straightedge) and a compass
(which draws circles) if and only if n is a power of
two times a product of distinct Fermat primes. Later Pierpont
[Pierpont1895] showed that
you can divide a circle into
n parts using origami (paper folding)
if and only if n is a product of a power of two times a power
of three times a distinct product of primes of the form
2n3m+1. These primes
are now called Pierpont primes.
Simple heuristics suggest that there should be finitely many Fermat primes, but infinitely many
Pierpont primes. In the following table we give a count of the numbers smaller Pierpont primes.
Circles can be
divided into the same numbers of parts using a
straight edge, compass
and an "angle trisector."
See Also: FermatNumber
Related pages (outside of this work)
- D. A. Cox and J. Shurman, "Geometry and number theory on clovers," Amer. Math. Monthly, 112:8 (2005) 682--704. MR2167769
- A. M. Gleason, "Angle trisection, the heptagon, and the triskaidecagon," Amer. Math. Monthly, 95:3 (1988) 185--194. MR935432
- R. K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, NY, 1994. ISBN 0-387-94289-0. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references. See his newer editions of this text.]
- G. E. Martin, Geometric constructions, Undergraduate Texts in Mathematics Springer-Verlag, New York, 1998. pp. xii+203, ISBN 0-387-98276-0. MR1483895
- J. Pierpont, "On an undemonstrated theorem of the Disquisitiones Aritmeticae," American Mathematical Society Bulletin,:2 (1895-1896) 77 - 83.