Pierpont prime


A Pierpont prime is a prime number of the form
for some nonnegative integers and. That is, they are the prime numbers for which 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 is of the form, and is therefore a Fermat prime. If is positive then must also be positive, and therefore the non-Fermat Piermont primes all have the form, when is a positive integer.
The first few Pierpont primes are:

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 condition that the exponent must be prime. Thus, it is expected that among -digit numbers of the correct form, 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 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 Pierpont primes up to. According to Gleason's conjecture there are Θ Pierpont primes smaller than N, as opposed to the smaller conjectural number O of Mersenne primes in that range.

Primality testing

When, the primality of 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 three.

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 gives values of m, k, and n such that
The left-hand side is a Pierpont prime when k is a power of 3; the right-hand side is a Fermat number.
mknYearDiscoverer
383411903Cullen, Cunningham & Western
639671956Robinson
20732091956Robinson
452274551956Robinson
9428994311983Keller
1218581121891993Dubner
2828181282851996Taura
15716731571691995Young
21331932133211996Young
30308833030931998Young
38244733824491999Cosgrave & Gallot
46107694610812003Nohara, Jobling, Woltman & Gallot
4957282434957322007Keiser, Jobling, Penné & Fougeron
672005276720072005Cooper, Jobling, Woltman & Gallot
2145351321453532003Cosgrave, Jobling, Woltman & Gallot
2478782324787852003Cosgrave, Jobling, Woltman & Gallot
2543548925435512011Brown, Reynolds, Penné & Fougeron

, the largest known Pierpont prime is 9·213334487 + 1, whose primality was discovered in March 2020.

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.
It follows that they allow any regular polygon of sides to be formed, as long as and 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, straightedge, and angle-trisector. Regular polygons which can be constructed with only compass and straightedge are the special case where and 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 -gons that can be constructed with these operations are the ones such that the totient 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 prime is 11; therefore, the hendecagon is the smallest regular polygon that cannot be constructed with compass, straightedge and angle trisector. All other regular -gons with can be constructed with compass, straightedge and trisector.

Generalization

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. The largest known Pierpont prime of the second kind that is not a Mersenne, is found by PrimeGrid.
A generalized Pierpont prime is a prime of the form with k fixed primes, pi < pj for i < j. A generalized Pierpont prime of the second kind is a prime of the form with k fixed primes, pi < pj 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:
+1−1