Pierpont prime

Pierpont prime
Named after	James Pierpont
No. of known terms	Thousands
Conjectured no. of terms	Infinite
Subsequence of	Pierpont number
First terms	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
Largest known term	3×210,829,346 + 1
OEIS index	A005109

A Pierpont prime is a prime number of the form

2^{u}3^{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, who introduced them in the study of regular polygons that can be constructed using conic sections.

A Pierpont prime with $v = 0$ is of the form $2^{u}+1$ , and is therefore a Fermat prime (unless $u = 0$ ). If $v$ is positive then $u$ must also be positive (because a number of the form $3^{v}+1$ would be even and therefore non-prime, since 2 cannot be expressed as $3^{v}+1$ when $v$ is a positive integer), and therefore the non-Fermat Piermont primes all have the form $6 k + 1$ , when $k$ is a positive integer (except for 2, when $u = v = 0$ ).

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 the OEIS)

Distribution

Unsolved problem in mathematics:

Are there infinitely many Pierpont primes?

(more unsolved problems in mathematics)

Distribution of the exponents for the smaller Pierpont primes

Empirically, the Pierpont primes do not seem to be particularly rare or sparsely distributed. There are 36 Pierpont primes less than 10⁶, 59 less than 10⁹, 151 less than 10²⁰, and 789 less than 10¹⁰⁰. There are few restrictions from algebraic factorisations on the Pierpont primes, so there are no requirements like the Mersenne prime condition that the exponent must be prime. Thus, it is expected that among $n$ -digit numbers of the correct form $2^{u}3^{v}+1$ , the fraction of these that are prime should be proportional to $1/ n$ , a similar proportion as the proportion of prime numbers among all $n$ -digit numbers. As there are $Θ (n 2)$ numbers of the correct form in this range, there should be $Θ (n)$ Pierpont primes.

Andrew M. Gleason made this reasoning explicit, conjecturing there are infinitely many Pierpont primes, and more specifically that there should be approximately $9 n$ Pierpont primes up to $10 n$ .^[1] According to Gleason's conjecture there are Θ(log N) Pierpont primes smaller than N, as opposed to the smaller conjectural number O(log log N) of Mersenne primes in that range.

Primality testing

When $2^{u}>3^{v}$ , the primality of $2^{u}3^{v}+1$ can be tested by Proth's theorem. On the other hand, when $2^{u}<3^{v}$ alternative primality tests for $p=2^{u}3^{v}+1$ are possible based on the factorization of $p-1$ as a small even number multiplied by a large power of three.^[2]

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^[3] 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 2017, the largest known Pierpont prime is 3 × 2^10829346 + 1,^[4] whose primality was discovered by Sai Yik Tang and PrimeGrid in 2014.^[5]

Polygon construction

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 the construction of the points that solve any cubic equation.^[6] It follows that they allow any regular polygon of $N$ sides to be formed, as long as $N \geq 3$ 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.^[1] 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.

In 1895, James Pierpont studied the same class of regular polygons; his work is what gives the name to the Pierpont primes. Pierpont generalized compass and straightedge constructions in a different way, by adding the ability to draw conic sections whose coefficients come from previously constructed points. As he showed, the regular $N$ -gons that can be constructed with these operations are the ones such that the totient of $N$ is 3-smooth. Since the totient of a prime is formed by subtracting one from it, the primes $N$ for which Pierpont's construction works are exactly the Pierpont primes. However, Pierpont did not describe the form of the composite numbers with 3-smooth totients.^[7] As Gleason later showed, these numbers are exactly the ones of the form $2 m 3 n ρ$ given above.^[1]

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 (or origami, or conic sections). All other regular $N$ -gons with $3 \leq N \leq 21$ can be constructed with compass, straightedge and trisector.^[1]

Generalization

A Pierpont prime of the second kind is a prime number of the form 2^u3^v − 1. These numbers 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, 131071, 139967, 165887, 294911, 314927, 442367, 472391, 497663, 524287, 786431, 995327, ... (sequence A005105 in the 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, ..., p_k}	+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 2 3 4 Gleason, Andrew M. (1988), "Angle trisection, the heptagon, and the triskaidecagon", American Mathematical Monthly, 95 (3): 185–194, doi:10.2307/2323624, MR 0935432 . Footnote 8, p. 191.
↑ Kirfel, Christoph; Rødseth, Øystein J. (2001), "On the primality of $2h\cdot 3^{n}+1$ ", Discrete Mathematics, 241 (1–3): 395–406, doi:10.1016/S0012-365X(01)00125-X, MR 1861431 .
↑ Wilfrid Keller, Fermat factoring status.
↑ Caldwell, Chris. "The largest known primes". The Prime Pages. Retrieved 14 March 2017.
↑ "PrimeGrid's 321 Prime Search" (PDF). PrimeGrid. Retrieved 14 March 2017.
↑ Hull, Thomas C. (2011), "Solving cubics with creases: the work of Beloch and Lill", American Mathematical Monthly, 118 (4): 307–315, doi:10.4169/amer.math.monthly.118.04.307, MR 2800341 .
↑ Pierpont, James (1895), "On an undemonstrated theorem of the Disquisitiones Arithmeticæ", Bulletin of the American Mathematical Society, 2 (3): 77–83, doi:10.1090/S0002-9904-1895-00317-1, MR 1557414 .

References

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

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[g98-1] 1 2 3 4 Gleason, Andrew M. (1988), "Angle trisection, the heptagon, and the triskaidecagon", American Mathematical Monthly, 95 (3): 185–194, doi:10.2307/2323624, MR 0935432 . Footnote 8, p. 191.

[2] Kirfel, Christoph; Rødseth, Øystein J. (2001), "On the primality of $2h\cdot 3^{n}+1$ ", Discrete Mathematics, 241 (1–3): 395–406, doi:10.1016/S0012-365X(01)00125-X, MR 1861431 .

[3] Wilfrid Keller, Fermat factoring status.

[4] Caldwell, Chris. "The largest known primes". The Prime Pages. Retrieved 14 March 2017.

[5] "PrimeGrid's 321 Prime Search" (PDF). PrimeGrid. Retrieved 14 March 2017.

[6] Hull, Thomas C. (2011), "Solving cubics with creases: the work of Beloch and Lill", American Mathematical Monthly, 118 (4): 307–315, doi:10.4169/amer.math.monthly.118.04.307, MR 2800341 .

[7] Pierpont, James (1895), "On an undemonstrated theorem of the Disquisitiones Arithmeticæ", Bulletin of the American Mathematical Society, 2 (3): 77–83, doi:10.1090/S0002-9904-1895-00317-1, MR 1557414 .

Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
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Base-dependent	Happy Dihedral Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k−Tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Safe ( $p, (p - 1)/2$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
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