number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, a Pierpont prime is a

prime number A prime number (or a prime) 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 the form

2^u\cdot 3^v + 1\,

for some nonnegative

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who used them to characterize the

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

s that can be constructed using

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...

s. The same characterization applies to polygons that can be constructed using ruler, compass, and

angle trisector 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 a ...

, or using

paper folding ) is the Japanese art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a fi ...

. Except for 2 and the

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 4294967 ...

s, 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

A Pierpont prime with is of the form

2^u+1

, and is therefore a

(unless ). If is

positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posit ...

then must also be positive (because

3^v+1

would be an

even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...

greater than 2 and therefore not prime), and therefore the non-Fermat Piermont primes all have the form , when is a positive integer (except for 2, when ). Pierpont exponent distribution

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 In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...

condition that the exponent must be prime. Thus, it is expected that among -digit numbers of the correct form

2^u\cdot3^v+1

, the fraction of these that are prime should be proportional to , a similar proportion as the proportion of prime numbers among all -digit numbers. As there are

\Theta(n^)

numbers of the correct form in this range, there should be

\Theta(n)

Pierpont primes.

Andrew M. Gleason Andrew Mattei Gleason (19212008) was an American mathematician who made fundamental contributions to widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in teaching at ...

made this reasoning explicit, conjecturing there are infinitely many Pierpont primes, and more specifically that there should be approximately Pierpont primes up to .. Footnote 8, p. 191. According to Gleason's conjecture there are

\Theta(\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

2^u\cdot 3^v + 1

is a

Proth number A Proth number is a natural number ''N'' of the form N = k \times 2^n +1 where ''k'' and ''n'' are positive integers, ''k'' is odd and 2^n > k. A Proth prime is a Proth number that is prime. They are named after the French mathematician François ...

and thus its primality can be tested by

Proth's theorem In number theory, Proth's theorem is a primality test for Proth numbers. It states that if ''p'' is a Proth number, of the form ''k''2''n'' + 1 with ''k'' odd and ''k'' < 2^''n'', and if there exists an

. On the other hand, when

2^u < 3^v

alternative primality tests for

M=2^u\cdot 3^v + 1

are possible based on the factorization of

M-1

as a small even number multiplied by a large

power of 3 In mathematics, a power of three is a number of the form where is an integer – that is, the result of exponentiation with number three as the base and integer as the exponent. Applications The powers of three give the place values i ...

Pierpont primes found as factors of Fermat numbers

As part of the ongoing worldwide search for factors of

Fermat number In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 4294967 ...

s, some Pierpont primes have been announced as factors. The following table 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. , the largest known Pierpont prime is 3 × 2^16408818 + 1 (4,939,547 decimal digits), whose primality was discovered in October 2020.

Polygon construction

In the

mathematics of paper folding The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper f ...

, 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 In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the ...

. It follows that they allow any

of sides to be formed, as long as and is of the form , where is a product of distinct Pierpont primes. This is the same class of regular polygons as those that can be constructed with a

compass A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with ...

straightedge A straightedge or straight edge is a tool used for drawing straight lines, or checking their straightness. If it has equally spaced markings along its length, it is usually called a ruler. Straightedges are used in the automotive service and ma ...

, and

. Regular polygons which can be constructed with only compass and straightedge (

constructible polygon 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 infinite ...

s) are the special case where and is a product of distinct

s, 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

s whose coefficients come from previously constructed points. As he showed, the regular -gons that can be constructed with these operations are the ones such that the

totient In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...

of is 3-smooth. Since the totient of a prime is formed by subtracting one from it, the primes 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. As Gleason later showed, these numbers are exactly the ones of the form given above. The smallest prime that is not a Pierpont (or Fermat) prime is 11; therefore, the

hendecagon In geometry, a hendecagon (also undecagon or endecagon) or 11-gon is an eleven-sided polygon. (The name ''hendecagon'', from Greek ''hendeka'' "eleven" and ''–gon'' "corner", is often preferred to the hybrid ''undecagon'', whose first part is f ...

is the first regular polygon that cannot be constructed with compass, straightedge and angle trisector (or origami, or conic sections). All other regular with can be constructed with compass, straightedge and trisector.

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

s; currently the largest known is

2^-1

(24,862,048 decimal digits). The largest known Pierpont prime of the second kind that is not a Mersenne prime is

3\cdot 2^-1

, found by

PrimeGrid PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing ...

.3*2^11895718 - 1
(3,580,969 Decimal Digits), from The

Prime Pages The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" ...

. A generalized Pierpont prime is a prime of the form

p_1^ \!\cdot p_2^ \!\cdot p_3^ \!\cdot \ldots \cdot p_k^ + 1

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

p_1^ \!\cdot p_2^ \!\cdot p_3^ \!\cdot \ldots \cdot p_k^ - 1

with ''k'' fixed primes ''p''₁ < ''p''₂ < ''p''₃ < ... < ''p''_''k''. Since all primes greater than 2 are

odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...

, in both kinds ''p''₁ must be 2. The sequences of such primes in the

OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...

are:

References