Pierpont prime

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 introduced them in the study of regular polygons that can be constructed using conic sections.

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:

Empirically, the Pierpont primes do not seem to be particularly rare or sparsely distributed. There are 36 Pierpont primes less than 10⁶, 59 less than 10⁹, 151 less than 10²⁰, and 789 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 $n$ -digit numbers of the correct form $2^{u}3^{v}+1$ , the fraction of these that are prime should be proportional to $1/ n$ , a similar proportion as the proportion of prime numbers among all $n$ -digit numbers. As there are $Θ(n 2)$ numbers of the correct form in this range, there should be $Θ(n)$ Pierpont primes.

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