# 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 2

^{n}3

^{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.

Circles can be divided into the same numbers of parts using a straight edge, compass and an "angle trisector."

NPierpont primes

belowN10 4 100 10 10,000 25 100,000,000 57 10 ^{16}125 10 ^{32}250 10 ^{64}505 10 ^{128}1020 10 ^{256}2075 10 ^{512}4227

**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. CoxandJ. 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, 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.]- 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.

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