## Re: [PrimeNumbers] Proving a smooth number +/- 1 prime

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• ... The Prime Pages contain all sorts of useful knowledge about prime numbers. For your question, yes, it s true, it can be established with certainty, and
Message 1 of 5 , Nov 1, 2005
ed pegg wrote:

> Is the following true?
>
> For any number n with less than 20000 digits, if n+1 or n-1 is
> an easily factorable smooth number, then the primality/non-primality
> of n can be established with certainty.
>
> If so, what is the primality proof method called?
>
> Ed Pegg Jr.
>
>
>
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
>
The Prime Pages contain all sorts of useful knowledge about prime
numbers. For your question, yes, it's true, it can be established with
certainty, and it's not just limited to 20,000 digits. In fact, by
modifying the tests, you don't have to completely factor 'N-1' or 'N+1'
... you just have to factor them 'enough'. See
tests and the 'N+1' tests. These pages will also give you references to
find more detailed information.

Jonathan Zylstra

>
>
> Mathematics education
> Mathematics and computer science
>
>
>
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• ... If you want old contents of a URL then try the Internet Archive: http://www.archive.org http://web.archive.org/web/20040922151554/http://powersum.dhis.org/
Message 2 of 5 , Nov 1, 2005
Phil Carmody wrote:

> Apparently I have the most complete mirror of Nuutti's site, and
> the most up-to-date information, but unfortunately some of it is
> ranges that is more up-to-date than the info at:
> http://fatphil.org/Nuutti/
> http://fatphil.org/Nuutti/minus/minus.html

If you want old contents of a URL then try the Internet Archive:
http://www.archive.org

http://web.archive.org/web/20040922151554/http://powersum.dhis.org/

--
Jens Kruse Andersen
• ... Yes, this and more is true. Any number n can be proven prime/composite easily , in time O(d^2 * log d * log log d) where d = log n, _if_ enough of the
Message 3 of 5 , Nov 1, 2005
Ed Pegg Jr. wrote:

> Is the following true?
>
> For any number n with less than 20000 digits, if n+1 or n-1 is
> an easily factorable smooth number, then the primality/non-primality
> of n can be established with certainty.

Yes, this and more is true. Any number n can be proven prime/composite
"easily", in time O(d^2 * log d * log log d) where d = log n, _if_ enough of
the prime factorization of n+1 or n-1 is known.
This time is much faster than the average time to find a probable prime.
The known prime factors of n+/-1 don't have to be small, they just have to be
proven primes.

> If so, what is the primality proof method called?

There are different proof methods with different names, depending on the form
and factorization percentage of n+/-1. The methods are generally called
"classical tests".
See the Prime Pages for details: http://primes.utm.edu/prove/prove3.html

The popular flexible program PrimeForm/GW implements a BLS proof
(Brillhart-Lehmer-Selfridge).
PrimeForm/GW can prove any n up to a million or more digits, if the product of
known factors of n-1 or n+1 is at least n^(1/3), or a little less for combined
n-1 and n+1 proofs.

Some BLS "extensions" not implemented in PrimeForm can prove numbers with less
n+/-1 factorization.
David Broadhurst's KP (Konyagin-Pomerance) PARI script can handle n^0.3.
John Renze's CHG (Coppersmith-Howgrave-Graham) PARI script (currently being
debugged) can handle down to around n^0.27, depending on the size of n, the
used computer, and the acceptable time.

It is utterly hopeless to find enough n+/-1 factorization to easily prove
primality of a random n above 1000 digits.
If you get "lucky", you can find a large probable prime cofactor of n-1 or
n+1. You cannot prove primality of that cofactor significantly easier than of
n itself, so it doesn't help in practice.

The largest proven prime n without large n+/-1 factorization is the ECPP
record with 15071 digits: http://primes.utm.edu/top20/page.php?id=27

The top-20 for a publicly available program is all with the ECPP program Primo
with record at 7993 digits: http://www.ellipsa.net/primo/top20.html

--
Jens Kruse Andersen
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