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
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