Pierpont prime

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.

The first few Pierpont primes are: It has been conjectured that there are infinitely many Pierpont primes, but this remains unproven.

A Pierpont prime with v = 0 is of the form

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.

Thus, it is expected that among n-digit numbers of the correct form

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

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.

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.

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.

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.

These numbers are The largest known primes of this type are Mersenne primes; currently the largest known is

with k fixed primes p1 < p2 < p3 < ... < pk.

Since all primes greater than 2 are odd, in both kinds p1 must be 2.