## Some Interesting Numbers – Proth Numbers, Rough Numbers and Pi-Primes [Part 2]

This is my second post to the Interesting Numbers series, where I’ll post a few types of numbers which I find interesting. In the last post I looked at Polygonal Numbers, Kaprekar’s Constant and Highly Composite Numbers. In this post, we’ll explore Proth Primes/Numbers, Rough Numbers and Pi-Primes. I’m going to try and improve upon my previous post, by adding some background history and more theorems if I can.

Proth Numbers:

Proth was a self taught Mathematician working as a farmer.

A Proth Number is any number of the following form, and satisfies the following condition:

$k \cdot 2^n + 1$,

where k is a positive odd integer and $2^n > k$

A Proth Prime is any Proth Number which additionally satisfies the condition of being prime.

Proth’s Theorem (1878) states that any Proth Number is also prime, if it satisfies the following condition:

$a^\frac{p-1}{2} \equiv -1 (mod \hspace{0.5mm}p)$, with a being some integer and p being a Proth Number.

OEIS Sequence A080075 (A080076 for Proth Primes) illustrates these numbers. The first Proth Numbers are as follows:

3, 5, 9, 17, 25, 33, 41, 49, 57, 65, 81, 97

Rough Numbers:

A k-rough number is a number whose prime factors are all greater than or equal to k. (Finch 2001/2003). Furthermore, Knuth and Greene added a further definition, which created unusual numbers. These numbers have the following a definition:

A number n is unusual if its greatest prime factor is greater than or equal to $\sqrt {n}$.

The probability of find such a number for some n is $ln \hspace {0.5mm} 2 \approx 0.6931471806$

It also follows that any number which is not k-rough, must then be k-smooth. Otherwise, if a number is prime, then the number is both rough and smooth, since no definition is  dominating.

Pi-Prime:

A number for all the Pi-enthusiasts, who love the hidden interesting numbers of Pi itself. A Pi-Prime, or more appropriately $\pi$-Prime, is any prime which forms part of the decimal expansion of the constant $\pi$. See OEIS A005042 for more details.

Additional, and more interesting versions are the subsequent floor and ceiling Pi-Primes.

$\lceil{\pi^n}\rceil$ and $\lfloor{\pi^n}\rfloor$, where n is some positive integer.

References:

Rough Number

Pi-Prime