Proth-like Numbers

Today's tweet for my numbered days was as follows:

It turns out that this number is part of a class of numbers of the form $k\times 2^n-1$ where $k$ is any odd integer and $n$ is a natural number. Here is a link to a website that shows values of $k$ between 1 and 299 and lists some corresponding values of $n$ that produce prime numbers. Note the site was updated on February 18th 2021.

This information can then be used to easily generate a very large prime number. For example, for $k=7$ some initial values of $n$ are:

1, 5, 9, 17, 21, 29, 45, 177, 18381, 22529, 24557, 26109, 34857, 41957, 67421, 70209, 169085, 173489, 177977, 363929, 372897

The prime number generated from $k\times 2^n-1$ when $k=7$ and $n=24557$ has 7394 decimal digits. In the case of $k=1$, the primes generated are Mersenne primes and $n$ itself must be a prime number. However, for larger values of $k$, $n$ does not need to be prime. Here is the list provided for $k=1$ at the previously mentioned site:

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951

Related to numbers of the form $k\times 2^n-1$ are the Proth numbers that are of the form $k\times 2^n+1$ and that I've written about in a blog post on January 18th 2020.

on Saturday, April 24th 2021