Pierpont prime

A Pierpont prime is a prime of the form. With the help of Pierpont primes may select specific which regular polygons can be constructed with ruler and compass as well as a tool for angle trisection. They are named after the American mathematician James Pierpont.

Definition

A prime number is called Pierpont - prime if it of the form

, said natural numbers. The Pierpont primes are thus those primes, is for the 3- smooth.

Examples

The first Pierpont primes are:

The currently largest known prime number Pierpont -

With over two million decimal places. Your primality was proved by Michael Herder, 2011.

Properties

Special cases

• For and there is no Pierpont primes, as is an even number greater than two and assembled therewith.
• For and a power of two and must be a Pierpont prime is thus a Fermat prime.
• For Piermont and a prime number has the form.

Distribution

The number of Pierpont primes is less than

The number of Pierpont primes is less than

Andrew Gleason conjectured that there are infinitely many primes Pierpont. They are not particularly rare and have few restrictions on algebraic factorizations. For example, there are no conditions, such as Mersenne primes, that the exponent must be prime. Presumably there are

Pierpont primes less than, in contrast to Mersenne primes in the same range.

Factors of Fermat numbers

As part of ongoing global search for factors of Fermat numbers, some Pierpont primes have been found to be factors. The following table gives values ​​for, and, so that:

The left side is a prime Pierpont, if a power of three; the right side is a Fermat number.

Applications

A regular polygon with sides can be accurately then constructed with ruler and compass as well as a tool for angle trisection, if the form of the

Is, being greater than three with different Pierpont primes. The constructible polygons, ie, polygons that can be constructed with ruler and compass, thereof are special cases in which are various and Fermat primes. The smallest prime that is not a Pierpont prime number is. Therefore, the Elfeck is the smallest regular polygon that can not be constructed with compass, ruler and angle trisection. All other regular corner with can be constructed a tool for angle trisection with compass, ruler and (where appropriate).

In the mathematics of paper folding the Huzita axioms define six of the seven possible folds. These folds are sufficient also to make each regular polygon with sides, if it is of the above form.