Monday, 27 June 2016

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×2n1 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×2n1 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 kn 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×2n1 are the Proth numbers that are of the form k×2n+1 and that I've written about in a blog post on January 18th 2020.

on Saturday, April 24th 2021

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