Verified primes of the form h*2^n-1 where h < 2^n
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README.md
h-0mod3-n.verified-prime.txt
h-n.verified-prime.txt
h-not0mod3-n.verified-prime.txt

README.md

verified-prime

The file, "h-n.verified-prime.txt":

https://github.com/arcetri/verified-prime/blob/master/h-n.verified-prime.txt

contains a list of verified primes of the form h*2n-1 where h < 2n and that have been proven to be prime by at least three or more independent tests.

An independent test means that proof of primality was performed on using a different code base. Moreover, each independent test was run on different hardware from the other tests.

Not only did we perform at least two different tests on these primes, others contributed their tests and helped further verify the primality of the values found in this list.

Primes were converted into canonical form where h is an odd value. Where the original list value had h even, n was incremented and h halved until h was odd.

As of this release, "h-n.verified-prime.txt" contains 146553 unique verified primes.

The file h-0mod3-n.verified-prime.txt:

https://github.com/arcetri/verified-prime/blob/master/h-0mod3-n.verified-prime.txt

is a subset of "h-n.verified-prime.txt" containing verified primes where h is a multiple of 3.

The file h-not0mod3-n.verified-prime.txt:

https://github.com/arcetri/verified-prime/blob/master/h-not0mod3-n.verified-prime.txt

is a subset of "h-n.verified-prime.txt" containing verified primes where h is a not multiple of 3.

v1-data

The v1-data subdirectory:

https://github.com/arcetri/verified-prime/tree/master/v1-data

contains information about a survey of v(1) values used for testing if a large integers of the form h*2n-1 are prime.

Details on this survey may be found in:

https://github.com/arcetri/verified-prime/tree/master/v1-data/README.md

Motivation

The main motivation for this list is to provide those developing code to search for new prime numbers with a large list of known prime numbers.

When we were testing GoPrime (see the https://github.com/arcetri/GoPrime site), we collected lists of primes of the form h*2n-1 and tested our code against those lists. To our surprise we discovered some of these prime lists contained multiple values that were not prime. In some cases the those lists we found contained values where primes as small as 3 were a factor of the number!

The v1-data subdirectory was created to support finding an optimal strategy for testing if a large integers of the form h*2n-1 are prime.

Usage

The file, "h-n.verified-prime.txt" contains a list of primes of the form h*2n-1, one per line. Each line contains a h value, followed by s space, followed by an n value followed by a newline. The values are written in base 10 (decimal).

We only considered h*2n-1 primes where h < 2n. For example, 289*23-1 is not found on the list, even though 2311 is prime because 289 >= 23.

Most primality tests for numbers of the form h*2n-1 require h < 2n and so we placed the same restriction on this list.

The files "h-0mod3-n.verified-prime.txt" and "h-not0mod3-n.verified-prime.txt" have the same format as "h-n.verified-prime.txt".

Example

The line:

1111 11

indicates that 1111*211-1 is prime. 1111*211-1 == 2275327 is a prime number.

Important NOTE

This list is NOT intended to be exhaustive. There are many primes of the form h*2n-1 that are NOT found in the list. There are many primes of the form h*2n-1 that we skipped.

Contribute

We went to extraordinary effort to make sure that the list contains only primes form h*2n-1 where h < 2n. While we would find it very surprising, nevertheless it is possible that the list contains mistakes.

If you discover that one of the values in our list is not prime, let us know by giving us a pull request to remove the non-prime from our list. Your pull request must include a note that either gives us factors of the numbers being removed or gives us a rigorous mathematical proof that the numbers removed are not prime.

Please do not

We are not very interested in expanding this list. Adding more primes to our list will not significantly improve the usefulness of these test cases.

Please do not send us primes to add to the list. It takes a lot of work to independently verify primes (more than once). The effort to add additional primes and to verify them is just not worth it. Sorry.

Please do not ask us to verify primes that you may have found. We do not have the time, nor the resources, to verify your primes. Sorry.

Please do not tell is about your theory of how to find prime numbers. We do not have the time, nor the resources, to look into new methods of finding primes. Sorry.

License

This project is distributed under the terms of the Apache License Version 2.0, January 2004. See file "LICENSE" for further reference.

If you do use this list, we would appreciate if you would tell us and/or mention in papers that you used our list.