Pierpont prime
From Wikipedia, the free encyclopedia
A Pierpont prime is a prime number greater than 3 having the form
for some integers u,v ≥ 0.
It is possible to prove that if v is zero, then u must be a power of 2, making p a Fermat prime.
The first few Pierpont primes are:
Pierpont primes are not particularly rare, and it is not difficult to locate moderately large ones. These are a few:
- 218 317 + 1 = 33853318889473
- 2 · 330 + 1 = 411782264189299
- 211 324 + 1 = 578415690713089
- 28 327 + 1 = 1952152956156673
- 210 327 + 1 = 7808611824626689
- 247 362 + 1 = 53694226297143959644031344050777763036004353
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
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 
, 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 2k3lp, where p is any Pierpont prime.
