Langmuir (unit)

2013-08-02T13:43:10Z

128.40.76.3: /* Definition */

A '''Pierpont prime''' is a [[prime number]] of the form <big>     2''u''3''v'' + 1</big> for some nonnegative [[integer]]s ''u'' and ''v''. That is, they are the prime numbers ''p'' for which ''p'' − 1 is [[Smooth number|3-smooth]]. They are named after the mathematician [[James Pierpont (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 number|positive]] then ''u'' must also be positive, and the Pierpont prime is of the form 6''k'' + 1 (because if ''u'' = 0 and ''v'' > 0 then 2''u''3''v'' + 1 is an even number greater than 2 and therefore composite).

The first few Pierpont primes are:

:[[2 (number)|2]], [[3 (number)|3]], [[5 (number)|5]], [[7 (number)|7]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[37 (number)|37]], [[73 (number)|73]], [[97 (number)|97]], [[109 (number)|109]], [[163 (number)|163]], [[193 (number)|193]], [[257 (number)|257]], [[433 (number)|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. {{OEIS|id=A005109}}

== Distribution of Pierpont primes ==
[[Image:Pierpont_exponent_distribution.png|thumb|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 number]]s, some Pierpont primes have been announced as factors. The following table<ref>Wilfrid Keller, [http://www.prothsearch.net/fermat.html Fermat factoring status].</ref> gives values of ''m'', ''k'', and ''n'' such that

:<math>k\cdot 2^n + 1\text{ divides }2^{2^m} + 1. \, </math>

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

{| class="wikitable"
|+
!''m'' !!''k''!! ''n'' !! Year !! Discoverer
|-
|38 || 3 || 41 || 1903 || [[James Cullen (mathematician)|Cullen]], [[Allan Joseph Champneys Cunningham|Cunningham]] & Western
|-
|63 || 9 || 67 || 1956 || [[Raphael M. Robinson|Robinson]]
|-
|207 || 3 || 209 || 1956 || Robinson
|-
|452 || 27 || 455 || 1956 || Robinson
|-
|9428 || 9 || 9431 || 1983 || Keller
|-
|12185 || 81 || 12189 || 1993 || [[Harvey Dubner|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 || [[John B. Cosgrave|Cosgrave]] & Gallot
|-
|461076 || 9 || 461081 || 2003 || Nohara, Jobling, [[George Woltman|Woltman]] & Gallot
|-
|672005 || 27 || 672007 || 2005 || [[Curtis Cooper (mathematician)|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 × 27033641 + 1,<ref>Chris Caldwell, [http://primes.utm.edu/primes/lists/short.txt The largest known primes] at The [[Prime Pages]].</ref> whose primality was discovered by Michael Herder in 2011.

In the [[mathematics of paper folding]], the [[Huzita–Hatori axioms|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''m''3''n''ρ, 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 polygon]]s) are the special case where ''n'' = 0 and ρ is a product of distinct [[Fermat prime]]s, 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).

== Notes ==
{{Reflist}}

== References ==
*{{MathWorld|title=Pierpont Prime|urlname=PierpontPrime}}

{{Prime number classes|state=collapsed}}

[[Category:Classes of prime numbers]]

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