Pierpont prime

A Pierpont prime is a prime number of the form

:$2^u\; 3^v\; +\; 1,$

for some nonnegative integers "u" and "v". They are named after the mathematician James Pierpont.

It is possible to prove that if "v" = 0 and "u" > 0, then "u" must be a power of 2, making the prime a Fermat prime. If "v" is positive then "u" must also be positive, and the Pierpont prime is of the form 6"k" + 1 (because if "u" = 0 and "v" > 0 then $2^u\; 3^v\; +\; 1$ is an even number greater than 2 and therefore composite).

The first few Pierpont primes are:

:2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769. OEIS|id=A005109

Distribution of Pierpont primes

Andrew Gleason conjectured there are infinitely many Pierpont primes. They are not particularly rare and there are few restrictions from algebraic factorisations, so there are no requirements like the Mersenne prime condition that the exponent must be prime. There are 36 Pierpont primes less than 10⁶, 59 less than 10⁹, 151 less than 10²⁰, and 789 less than 10¹⁰⁰; conjecturally there are $O(log\; N)$ Pierpont primes smaller than "N", as opposed to the conjectured $O(log\; log\; N)$ Mersenne primes in that range.

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 [Wilfrid Keller, [http://www.prothsearch.net/fermat.html Fermat factoring status] .] gives values of "m", "k", and "n" such that

:$kcdot\; 2^n\; +\; 1$ divides $2^\{2^m\}\; +\; 1.$

The left-hand side is a Pierpont prime when "k" is a power of 3; the right-hand side is a Fermat number.

As of 2008, the largest known Pierpont prime is 3 · 2^2478785 + 1, [Chris Caldwell, [http://primes.utm.edu/primes/lists/short.txt The largest known primes] at The Prime Pages.] whose primality was discovered by John B. Cosgrave in 2003 with software by Paul Jobling, George Woltman, and Yves Gallot. [ [http://primes.utm.edu/bios/code.php?code=g245 Proof-code: g245] at The Prime Pages.]

In the mathematics of paper folding, Huzita's axioms define six of the seven types of fold possible. It has been shown that these folds are sufficient to allow any regular polygon of "N" sides to be formed, as long as "N" > 3 and of the form 2^"m"3^"n"ρ, 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 ruler, straightedge, and angle-trisector. Regular polygons which can be constructed with only ruler 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.

Notes

References

*MathWorld|title=Pierpont Prime|urlname=PierpontPrime