Pierpont prime

A Pierpont prime is a prime number of the form
     2^u3^v + 1
for some nonnegative integers u and v. That is, they are the prime numbers p for which p − 1 is 3-smooth. 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, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329. (sequence A005109 in 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^[1] 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 2011, the largest known Pierpont prime is 3 × 2^7033641 + 1,^[2] whose primality was discovered by Michael Herder in 2011.

In the mathematics of paper folding, the Huzita 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 compass, straightedge, and angle-trisector. Regular polygons which can be constructed with only compass 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.

The smallest prime that is not a Pierpont (or Fermat) prime is 11; therefore, the hendecagon is the smallest regular polygon that cannot be constructed with compass, straightedge and angle trisector. All other regular n-gons with 3 ≤ n ≤ 21 can be constructed with compass, straightedge and trisector (if needed).

Generalization

A Pierpont prime of the second kind is a prime number of the form 2^u3^v − 1, they are

2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, 383, 431, 647, 863, 971, 1151, 2591, 4373, 6143, 6911, 8191, 8747, 13121, 15551, 23327, 27647, 62207, 73727, ... (sequence A005105 in OEIS)

A generalized Pierpont prime is a prime of the form $p_1^{n_1} \cdot p_2^{n_2} \cdot p_3^{n_3} \cdot ... \cdot p_k^{n_k} + 1$ with k fixed primes {p₁, p₂, p₃, ..., p_k}, p_i < p_j for i < j. A generalized Pierpont prime of the second kind is a prime of the form $p_1^{n_1} \cdot p_2^{n_2} \cdot p_3^{n_3} \cdot ... \cdot p_k^{n_k} - 1$ with k fixed primes {p₁, p₂, p₃, ..., p_k}, p_i < p_j for i < j. Since all primes greater than 2 are odd, in both kinds p₁ must be 2. The sequences of such primes in OEIS are:

{p1, p2, p3, ..., pk}	+1	−1
{2}	A092506	A000668
{2, 3}	A005109	A005105
{2, 5}	A077497	A077313
{2, 3, 5}	A002200
{2, 7}	A077498	A077314
{2, 3, 5, 7}	A174144
{2, 11}	A077499	A077315
{2, 13}	A173236	A173062

Notes

1. Wilfrid Keller, Fermat factoring status.
2. Chris Caldwell, The largest known primes at The Prime Pages.

References

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