Pierpont prime

A Pierpont prime is a prime number greater than 3 having the form

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

for some integers u,v ≥ 0. They are named after James Pierpont.

It is possible to prove that if v is zero, then u must be a power of 2, making the prime a Fermat prime.

The first few Pierpont primes are:

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. In other words, they are not particularly rare—in particular, 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

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

The left-hand side is a Pierpont prime; the right-hand side is a Fermat number.[1]

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 2005, the largest known Pierpont prime is $2^{2478785}\cdot 3+1$, whose primality was discovered by John Cosgrave, Paul Jobling, George Woltman, and Yves Gallot in 2003.[2][3]

In the mathematics of paper folding, Huzita's 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 is 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.