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 × 220,498,148 + 1 |
OEIS index | A005109 |
In number theory, a Pierpont prime is a prime number of the form
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 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.
Are there infinitely many Pierpont primes?
A Pierpont prime with v = 0 is of the form , and is therefore a Fermat prime (unless u = 0). If v is positive then u must also be positive (because would be an even number greater than 2 and therefore not prime), and therefore the non-Fermat 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 106, 65 less than 109, 157 less than 1020, and 795 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 , 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 numbers of the correct form in this range, there should be 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 10n. [1] According to Gleason's conjecture there are Pierpont primes smaller than N, as opposed to the smaller conjectural number of Mersenne primes in that range.
When , is a Proth number and thus its primality can be tested by Proth's theorem. On the other hand, when alternative primality tests for are possible based on the factorization of as a small even number multiplied by a large power of 3. [2]
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 left-hand side is a Fermat number; the right-hand 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 [update] , the largest known Pierpont prime is 81 × 220498148 + 1 (6,170,560 decimal digits), whose primality was discovered in June 2023. [4]
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 2m3nρ, 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. [6] As Gleason later showed, these numbers are exactly the ones of the form 2m3nρ 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 N-gons with 3 ≤ N ≤ 21 can be constructed with compass, straightedge and trisector. [1]
A Pierpont prime of the second kind is a prime number of the form 2u3v − 1. These numbers are
The largest known primes of this type are Mersenne primes; currently the largest known is (24,862,048 decimal digits). The largest known Pierpont prime of the second kind that is not a Mersenne prime is , found by PrimeGrid. [7]
A generalized Pierpont prime is a prime of the form with k fixed primes p1 < p2 < p3 < ... < pk. A generalized Pierpont prime of the second kind is a prime of the form with k fixed primes p1 < p2 < p3 < ... < pk. Since all primes greater than 2 are odd, in both kinds p1 must be 2. The sequences of such primes in the OEIS are:
{p1, p2, p3, ..., pk} | + 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 |
In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length can be constructed with compass and straightedge in a finite number of steps. Equivalently, is constructible if and only if there is a closed-form expression for using only integers and the operations for addition, subtraction, multiplication, division, and square roots.
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures.
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.
In mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form
11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables.
The Huzita–Justin axioms or Huzita–Hatori axioms are a set of rules related to the mathematical principles of origami, describing the operations that can be made when folding a piece of paper. The axioms assume that the operations are completed on a plane, and that all folds are linear. These are not a minimal set of axioms but rather the complete set of possible single folds.
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.
In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon.
31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known.
In geometry, a hendecagon or 11-gon is an eleven-sided polygon.
In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon.
In geometry, the neusis is a geometric construction method that was used in antiquity by Greek mathematicians.
The number 4,294,967,295 is a whole number equal to 232 − 1. It is a perfect totient number, meaning it is equal to the sum of its iterated totients. It follows 4,294,967,294 and precedes 4,294,967,296. It has a factorization of .
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, the field of rational numbers.
In geometry, an icositrigon or 23-gon is a 23-sided polygon. The icositrigon has the distinction of being the smallest regular polygon that is not neusis constructible.
A Proth number is a natural number N of the form where k and n are positive integers, k is odd and . A Proth prime is a Proth number that is prime. They are named after the French mathematician François Proth. The first few Proth primes are
Geometric Origami is a book on the mathematics of paper folding, focusing on the ability to simulate and extend classical straightedge and compass constructions using origami. It was written by Austrian mathematician Robert Geretschläger and published by Arbelos Publishing in 2008. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.