Pierpont prime

A Pierpont prime is a prime number of the form

$2^{u}3^{v}+1\,$

for some nonnegative integers u and v. They are named after the mathematician James Pierpont.

It is possible to prove that if v = 0 and u > 0, then u must be a power of 2, making the prime a Fermat prime. If v is positive then u must also be positive, and the Pierpont prime is of the form 6k + 1 (because if u = 0 and v > 0 then 2^u3^{v + 1 is an even number greater than 2 and therefore composite).}

The first few Pierpont primes are:

2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769. (sequence A005109 in the OEIS)

Distribution of Pierpont primes

Distribution of the exponents for the smaller Pierpont primes

Andrew Gleason conjectured there are infinitely many Pierpont primes. They are not particularly rare and there are few restrictions from algebraic factorisations, so there are no requirements like the Mersenne prime condition that the exponent must be prime. There are 36 Pierpont primes less than 10⁶, 59 less than 10⁹, 151 less than 10²⁰, and 789 less than 10¹⁰⁰; conjecturally there are O(log N) Pierpont primes smaller than N, as opposed to the conjectured O(log log N) Mersenne primes in that range.

Pierpont primes found as factors of Fermat numbers

As part of the ongoing worldwide search for factors of Fermat numbers, some Pierpont primes have been announced as factors. The following table gives values of m, k, and n such that

$k\cdot 2^{n}+1{\text{ divides }}2^{2^{m}}+1.\,$

The left-hand side is a Pierpont prime when k is a power of 3; the right-hand side is a Fermat number.

m	k	n	Year	Discoverer
38	3	41	1903	Cullen, Cunningham & Western
63	9	67	1956	Robinson
207	3	209	1956	Robinson
452	27	455	1956	Robinson
9428	9	9431	1983	Keller
12185	81	12189	1993	Dubner
28281	81	28285	1996	Taura
157167	3	157169	1995	Young
213319	3	213321	1996	Young
303088	3	303093	1998	Young
382447	3	382449	1999	Cosgrave & Gallot
461076	9	461081	2003	Nohara, Jobling, Woltman & Gallot
672005	27	672007	2005	Cooper, Jobling, Woltman & Gallot
2145351	3	2145353	2003	Cosgrave, Jobling, Woltman & Gallot
2478782	3	2478785	2003	Cosgrave, Jobling, Woltman & Gallot

As of 2008, the largest known Pierpont prime is 3 · 2^2478785 + 1, whose primality was discovered by John B. Cosgrave in 2003 with software by Paul Jobling, George Woltman, and Yves Gallot.

In the mathematics of paper folding, the Huzita–Hatori axioms define six of the seven types of fold possible. It has been shown that these folds are sufficient to allow any regular polygon of N sides to be formed, as long as N > 3 and of the form 2^m3ⁿρ, where ρ is a product of distinct Pierpont primes. This is the same class of regular polygons as those that can be constructed with a ruler, straightedge, and angle-trisector. Regular polygons which can be constructed with only ruler and straightedge (constructible polygons) are the special case where n = 0 and ρ is a product of distinct Fermat primes, themselves a subset of Pierpont primes.

References

• Weisstein, Eric W. "Pierpont Prime". MathWorld.