Pierpont prime number

A Pierpont prime is a prime of the form . With the help of the Pierpont prime numbers, it is possible to specify which regular polygons can be constructed with compasses and rulers as well as an aid for angle trisection . They are named after the American mathematician James Pierpont . ${\displaystyle 2^{u}3^{v}+1}$

definition

A prime number is called a Pierpont prime number if it has the form ${\displaystyle p}$

${\displaystyle p=2^{u}3^{v}+1}$

is, where are natural numbers. The Pierpont prime numbers are thus those prime numbers for which 3-smooth is. ${\displaystyle u,v\in \mathbb {N} _{0}}$ ${\displaystyle p}$ ${\displaystyle p-1}$

Examples

The first Pierpont prime numbers 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, ... (Follow A005109 in OEIS )

The currently largest known Pierpont prime is

${\displaystyle 3\cdot 2^{10.829.346}+1}$

with 3,259,959 decimal places. Its primality was proven by Sai Yik Tang in 2014.

properties

Special cases

• There are no Pierpont prime numbers for and , because an even number is greater than two and thus a composite . ${\displaystyle u=0}$ ${\displaystyle v>0}$ ${\displaystyle 3^{v}+1}$
• For and must be a power of two and a Pierpont prime is therefore a Fermat prime number . ${\displaystyle u>0}$ ${\displaystyle v=0}$ ${\displaystyle u}$
• For and a Piermont prime has the form . ${\displaystyle u>0}$ ${\displaystyle v>0}$ ${\displaystyle 6k+1}$

distribution

Distribution of the exponents of the small Pierpont prime numbers

The number of Pierpont primes is less than is ${\displaystyle 10,100,1000,\ldots }$

${\displaystyle 4,10,18,25,32,42,50,58,65,72,78,83,93,106,114,125,139,\ldots }$   (Follow A113420 in OEIS ).

The number of Pierpont primes is less than is ${\displaystyle 10^{1},10^{2},10^{4},10^{8},\ldots }$

${\displaystyle 4,10,25,58,125,250,505,1020,2075,4227,8597,17213\ldots }$   (Follow A113412 in OEIS ).

Andrew Gleason hypothesized that there are an infinite number of Pierpont prime numbers. They are not particularly rare and have few restrictions on algebraic factoring. For example, there are no conditions, as with Mersenne prime numbers , that the exponent must be prime. Presumably there is

${\displaystyle O(\log N)}$

Pierpont prime numbers smaller than , in contrast to Mersenne prime numbers in the same range. ${\displaystyle N}$ ${\displaystyle O(\log \log N)}$

Factors of Fermat Numbers

As part of the ongoing worldwide search for Fermat number factors, some Pierpont prime numbers have already been found as factors. The following table gives values ​​for , and such that: ${\displaystyle m}$ ${\displaystyle k}$ ${\displaystyle n}$

${\displaystyle k\cdot 2^{n}+1{\text{ teilt }}2^{2^{m}}+1}$.

The left hand side is a Pierpont prime if it is a power of three; the right side is a Fermat number. ${\displaystyle k}$

${\displaystyle m}$	${\displaystyle k}$	${\displaystyle n}$	year	Explorer
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	basement, cellar
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
495728	243	495732	2007	Keizer, Jobling, Penné & others
672005	27	672007	2005	Cooper , Jobling, Woltman & Gallot
2145351	3	2145353	2003	Cosgrave, Jobling, Woltman & Gallot
2478782	3	2478785	2003	Cosgrave, Jobling, Woltman & Gallot
2543548	9	2543551	2011	Brown, Reynolds, Penné & Fougeron

Applications

A regular polygon with sides can be constructed with a pair of compasses and a ruler and a tool to divide the angles into three precisely if of the shape ${\displaystyle N}$ ${\displaystyle N}$

${\displaystyle N=2^{m}3^{n}p_{1}\cdots p_{k}}$

, wherein with various Pierpont primes greater than three. The polygons that can be constructed , i.e. the polygons that can only be constructed with compasses and ruler, are special cases in which and are different Fermat prime numbers. The smallest prime that is not a Pierpont prime is . Therefore, the elagon is the smallest regular polygon that cannot be constructed with a compass, ruler and angled thirds. All other regular corners with can be constructed with a compass, ruler and (if necessary) a tool to divide the angle into three parts. ${\displaystyle p_{1},\ldots ,p_{k}}$ ${\displaystyle k\in \mathbb {N} _{0}}$ ${\displaystyle n=0}$ ${\displaystyle p_{1},\ldots ,p_{k}}$ ${\displaystyle 11}$ ${\displaystyle n}$ ${\displaystyle 3\leq n\leq 21}$

In the mathematics of paper folding , the Huzita axioms define six of the seven possible folds. These folds are also sufficient to form any regular polygon with sides if of the above shape. ${\displaystyle N}$ ${\displaystyle N}$

generalization

A Pierpont prime of the 2nd kind is a prime of the form . The first numbers of this kind are: ${\displaystyle 2^{u}\cdot 3^{v}-1}$

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, ... (Follow A005105 in OEIS )

A generalized Pierpont prime is a prime of the form with k different, ever larger ordered prime numbers . ${\displaystyle p_{1}^{n_{1}}\cdot p_{2}^{n_{2}}\cdot p_{3}^{n_{3}}\cdot \ldots \cdot p_{k}^{n_{k}}+1}$ ${\displaystyle p_{1},p_{2},\ldots ,p_{k}}$

A generalized Pierpont prime number of the 2nd kind is a prime number of the form with k different, ever increasing ordered prime numbers . ${\displaystyle p_{1}^{n_{1}}\cdot p_{2}^{n_{2}}\cdot p_{3}^{n_{3}}\cdot \ldots \cdot p_{k}^{n_{k}}-1}$ ${\displaystyle p_{1},p_{2},\ldots ,p_{k}}$

In both cases it must be. All others are odd prime numbers. ${\displaystyle p_{1}=2}$ ${\displaystyle p_{i}}$

This previous statement results from the following consideration: If p _{1 were} not 2, the product of odd prime powers would again be odd. If you then add or subtract 1, the resulting number would definitely be even and therefore not prime. ${\displaystyle p_{1}^{n_{1}}\cdot p_{2}^{n_{2}}\cdot p_{3}^{n_{3}}\cdot \ldots \cdot p_{k}^{n_{k}}}$

Here are a few generalized Pierpont primes:

{p 1 , p 2 , p 3 ,…, p k }	+1	OEIS episode	-1	OEIS sequence
{2}	2, 3, 5, 17, 257, 65537	(Follow A092506 in OEIS )	3, 7, 31, 127, 8191, 131071, ...	(Follow A000668 in OEIS )
{2, 3}	2, 3, 5, 7, 13, 17, 19, 37, 73, 97, ...	(Follow A005109 in OEIS )	2, 3, 5, 7, 11, 17, 23, 31, 47, 53, ...	(Follow A005105 in OEIS )
{2, 5}	2, 3, 5, 11, 17, 41, 101, ...	(Follow A077497 in OEIS )	3, 7, 19, 31, 79, 127, 199, ...	(Follow A077313 in OEIS )
{2, 3, 5}	2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 41, ...	(Follow A002200 in OEIS )
{2, 7}	2, 3, 5, 17, 29, 113, 197, ...	(Follow A077498 in OEIS )	3, 7, 13, 31, 97, 127, 223, ...	(Follow A077314 in OEIS )
{2, 3, 5, 7}	2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, ...	(Follow A174144 in OEIS )
{2, 11}	2, 3, 5, 17, 23, 89, 257, 353, ...	(Follow A077499 in OEIS )	3, 7, 31, 43, 127, 241, 967, ...	(Follow A077315 in OEIS )
{2, 13}	2, 3, 5, 17, 53, 257, 677, ...	(Follow A173236 in OEIS )	3, 7, 31, 103, 127, 337, ...	(Follow A173062 in OEIS )

Web links

• Eric W. Weisstein : Pierpont Prime . In: MathWorld (English).
• Chris Caldwell: Pierpont prime. In: The Prime Pages. Retrieved May 16, 2013 . 

Individual evidence

1. ↑ Chris Caldwell: The largest known primes. In: The Prime Pages. August 16, 2016. Retrieved August 17, 2016 . 
2. ↑ Chris Caldwell: 3 210829346 + 1. In: The Prime Pages. January 17, 2014, accessed August 17, 2016 . 
3. ^ ^{A b} Andrew Gleason : Angle Trisection, the Heptagon, and the Triskaidecagon . In: The American Mathematical Monthly . tape 95 , no. 3 , 1988, pp. 185-194 ( math.nthu.edu.tw [PDF]). math.nthu.edu.tw ( Memento of the original from February 2, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
4. ^ Wilfrid Keller: Prime factors of Fermat numbers and complete factoring status. (No longer available online.) April 30, 2015, archived from the original on February 10, 2016 ; accessed on August 17, 2016 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
5. ↑ Follow A048135 in OEIS