Pierpont prime

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 2ⁿ3^m+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.

N	Pierpont primes below N
10	4
100	10
10,000	25
100,000,000	57
1016	125
1032	250
1064	505
10128	1020
10256	2075
10512	4227

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)

• A005109 Sloane Integer Sequence
• Pierpont Prime from Eric Weisstein's World of Mathematics

References:

CS2005

D. A. Cox and J. Shurman, "Geometry and number theory on clovers," Amer. Math. Monthly, 112:8 (2005) 682--704.  MR2167769

Gleason1988

A. M. Gleason, "Angle trisection, the heptagon, and the triskaidecagon," Amer. Math. Monthly, 95:3 (1988) 185--194.  MR935432

Guy94

R. K. Guy, Unsolved problems in number theory, Springer-Verlag, 1994.  New York, NY, 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.]

Martin1998

G. E. Martin, Geometric constructions, Undergraduate Texts in Mathematics Springer-Verlag, New York, 1998.  pp. xii+203, ISBN 0-387-98276-0. MR1483895

Pierpont1895

J. Pierpont, "On an undemonstrated theorem of the Disquisitiones Aritmeticae," American Mathematical Society Bulletin,:2 (1895-1896) 77 - 83.