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  81 × 2^{20,498,148} + 1 
OEIS index  A005109 
In number theory, a Pierpont prime is a prime number of the form [math]\displaystyle{ 2^u\cdot 3^v + 1\, }[/math] for some nonnegative integers u and v. That is, they are the prime numbers p for which p − 1 is 3smooth. They are named after the mathematician James Pierpont, who used them to characterize the regular polygons that can be constructed using conic sections. The same characterization applies to polygons that can be constructed using ruler, compass, and angle trisector, or using paper folding.
Except for 2 and the Fermat primes, every Pierpont prime must be 1 modulo 6. The first few Pierpont primes are:
It has been conjectured that there are infinitely many Pierpont primes, but this remains unproven.
Distribution
Unsolved problem in mathematics: Are there infinitely many Pierpont primes? (more unsolved problems in mathematics)

A Pierpont prime with v = 0 is of the form [math]\displaystyle{ 2^u+1 }[/math], and is therefore a Fermat prime (unless u = 0). If v is positive then u must also be positive (because [math]\displaystyle{ 3^v+1 }[/math] would be an even number greater than 2 and therefore not prime), and therefore the nonFermat Pierpont primes all have the form 6k + 1, when k is a positive integer (except for 2, when u = v = 0).
Empirically, the Pierpont primes do not seem to be particularly rare or sparsely distributed; there are 42 Pierpont primes less than 10^{6}, 65 less than 10^{9}, 157 less than 10^{20}, and 795 less than 10^{100}. 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 ndigit numbers of the correct form [math]\displaystyle{ 2^u\cdot3^v+1 }[/math], the fraction of these that are prime should be proportional to 1/n, a similar proportion as the proportion of prime numbers among all ndigit numbers. As there are [math]\displaystyle{ \Theta(n^{2}) }[/math] numbers of the correct form in this range, there should be [math]\displaystyle{ \Theta(n) }[/math] Pierpont primes.
Andrew M. Gleason made this reasoning explicit, conjecturing there are infinitely many Pierpont primes, and more specifically that there should be approximately 9n Pierpont primes up to 10^{n}.^{[1]} According to Gleason's conjecture there are [math]\displaystyle{ \Theta(\log N) }[/math] Pierpont primes smaller than N, as opposed to the smaller conjectural number [math]\displaystyle{ O(\log \log N) }[/math] of Mersenne primes in that range.
Primality testing
When [math]\displaystyle{ 2^u \gt 3^v }[/math], [math]\displaystyle{ 2^u\cdot 3^v + 1 }[/math] is a Proth number and thus its primality can be tested by Proth's theorem. On the other hand, when [math]\displaystyle{ 2^u \lt 3^v }[/math] alternative primality tests for [math]\displaystyle{ M=2^u\cdot 3^v + 1 }[/math] are possible based on the factorization of [math]\displaystyle{ M1 }[/math] as a small even number multiplied by a large power of 3.^{[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
The lefthand side is a Fermat number; the righthand side is a Pierpont prime.
m  k  n  Year  Discoverer 

38  1  41  1903  Cullen, Cunningham & Western 
63  2  67  1956  Robinson 
207  1  209  1956  Robinson 
452  3  455  1956  Robinson 
9428  2  9431  1983  Keller 
12185  4  12189  1993  Dubner 
28281  4  28285  1996  Taura 
157167  1  157169  1995  Young 
213319  1  213321  1996  Young 
303088  1  303093  1998  Young 
382447  1  382449  1999  Cosgrave & Gallot 
461076  1  461081  2003  Nohara, Jobling, Woltman & Gallot 
495728  5  495732  2007  Keiser, Jobling, Penné & Fougeron 
672005  3  672007  2005  Cooper, Jobling, Woltman & Gallot 
2145351  1  2145353  2003  Cosgrave, Jobling, Woltman & Gallot 
2478782  1  2478785  2003  Cosgrave, Jobling, Woltman & Gallot 
2543548  2  2543551  2011  Brown, Reynolds, Penné & Fougeron 
(As of 2023), the largest known Pierpont prime is 81 × 2^{20498148} + 1 (6,170,560 decimal digits), whose primality was discovered in June 2023.^{[4]}
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.^{[5]} It follows that they allow any regular polygon of N sides to be formed, as long as N ≥ 3 and is 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 Ngons that can be constructed with these operations are the ones such that the totient of N is 3smooth. 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 3smooth totients.^{[6]} 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 first regular polygon that cannot be constructed with compass, straightedge and angle trisector (or origami, or conic sections). All other regular Ngons with 3 ≤ N ≤ 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^{u}3^{v} − 1. These numbers are
The largest known primes of this type are Mersenne primes; currently the largest known is [math]\displaystyle{ 2^{82589933}1 }[/math] (24,862,048 decimal digits). The largest known Pierpont prime of the second kind that is not a Mersenne prime is [math]\displaystyle{ 3\cdot 2^{20928756}1 }[/math], found by PrimeGrid.^{[7]}
A generalized Pierpont prime is a prime of the form [math]\displaystyle{ p_1^{n_1} \!\cdot p_2^{n_2} \!\cdot p_3^{n_3} \!\cdot \ldots \cdot p_k^{n_k} + 1 }[/math] with k fixed primes p_{1} < p_{2} < p_{3} < ... < p_{k}. A generalized Pierpont prime of the second kind is a prime of the form [math]\displaystyle{ p_1^{n_1} \!\cdot p_2^{n_2} \!\cdot p_3^{n_3} \!\cdot \ldots \cdot p_k^{n_k}  1 }[/math] with k fixed primes p_{1} < p_{2} < p_{3} < ... < p_{k}. Since all primes greater than 2 are odd, in both kinds p_{1} must be 2. The sequences of such primes in the OEIS are:
{p_{1}, p_{2}, p_{3}, ..., p_{k}}  + 1  − 1 
{2}  OEIS: A092506  OEIS: A000668 
{2, 3}  OEIS: A005109  OEIS: A005105 
{2, 5}  OEIS: A077497  OEIS: A077313 
{2, 3, 5}  OEIS: A002200  OEIS: A293194 
{2, 7}  OEIS: A077498  OEIS: A077314 
{2, 3, 5, 7}  OEIS: A174144  OEIS: A347977 
{2, 11}  OEIS: A077499  OEIS: A077315 
{2, 13}  OEIS: A173236  OEIS: A173062 
See also
 Proth prime, the primes of the form [math]\displaystyle{ N = k \cdot 2^n + 1 }[/math] where k and n are positive integers, [math]\displaystyle{ k }[/math] is odd and [math]\displaystyle{ 2^n \gt k. }[/math]
References
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} "Angle trisection, the heptagon, and the triskaidecagon", American Mathematical Monthly 95 (3): 185–194, 1988, doi:10.2307/2323624. Footnote 8, p. 191.
 ↑ Kirfel, Christoph; Rødseth, Øystein J. (2001), "On the primality of [math]\displaystyle{ 2h\cdot 3^n+1 }[/math]", Discrete Mathematics 241 (13): 395–406, doi:10.1016/S0012365X(01)00125X.
 ↑ Wilfrid Keller, Fermat factoring status.
 ↑ Caldwell, Chris, The largest known primes, http://primes.utm.edu/primes/lists/short.txt, retrieved 17 June 2023; The Prime Database: 81*2^20498148+1, https://primes.utm.edu/primes/page.php?id=136165, retrieved 17 June 2023
 ↑ "Solving cubics with creases: the work of Beloch and Lill", American Mathematical Monthly 118 (4): 307–315, 2011, doi:10.4169/amer.math.monthly.118.04.307.
 ↑ "On an undemonstrated theorem of the Disquisitiones Arithmeticæ", Bulletin of the American Mathematical Society 2 (3): 77–83, 1895, doi:10.1090/S000299041895003171.
 ↑ 3*2^20928756  1 (6,300,184 Decimal Digits), from The Prime Pages.
Original source: https://en.wikipedia.org/wiki/Pierpont prime.
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