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Message boards : The Riesel Problem : How to prove?

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majortim

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Message 72940 - Posted: 31 Jan 2014 | 12:31:22 UTC

I have to admit I wasn't the best in maths in my school, so I wonder how can it be proven that a number is a Riesel number?
I mean as a Riesel number is a number k, which, when multiplied with 2^n and then 1 is subtracted yields no prime. But, as n can be any integer greater or equal 1, wouldn't you have to compute with an infinite amount of n?

Iain Bethune
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Message 72947 - Posted: 31 Jan 2014 | 13:51:05 UTC - in response to Message 72940.

Hi Majortim,

The original proof can be found in a 1958 paper by Hans Riesel (Riesel, Hans (1956). "NÃ¥gra stora primtal". Elementa 39: 258â€“260). It's in Swedish and I don't have my copy of his book to hand to give the details. However, the essential point is that he showed that for certain numbers e.g. k=509203

then all integers of the form k*2^n-1 are divisible by one (or more?) of the following small set of primes {3, 5, 7, 13, 17, 241}. This is known as a 'covering set' since it covers all values of n. Thus whatever (integer) n you choose, k*2^n-1 is always composite, so k=509203 is a Riesel number.

This method has not been successfully applied to any smaller k, so the 'Riesel Problem' is to prove that k=509203 is in fact the smallest Riesel number - by exhaustively testing all the smaller k to find an n which gives a prime. Currently there are 52 k remaining which could be a Riesel number (although we expect not!), and PrimeGrid is testing these.

If you haven't already been there, I recommend to check out http://www.prothsearch.com/rieselprob.html which gives some history, the latest status and some further links you might like to read.

Cheers

- Iain
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3073428256125*2^1290000-1 is Prime!

majortim

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Message 72950 - Posted: 31 Jan 2014 | 14:22:36 UTC

Thank you, Iain :-)
Hmm, it's not really a legimate question, but just out of curiousity, is there any practical use of Riesel numbers? (If not, I won't stop contributing ;-) )

So, if I (or rather my CPUs) find a Riesel number, will I get a message on my account or would I have to check some list here?

Iain Bethune
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Message 72959 - Posted: 31 Jan 2014 | 16:37:52 UTC - in response to Message 72950.

Is there any practical use of Riesel numbers?

Not that I know of!

If you happen to be lucky enough to find a prime in the TRP search, I believe you should get an email (unless your Project Prefs are set to prohibit this). In any case a TRP find is a very significant thing, and would be a Mega-prime, so there will be a post in the News forum and no doubt some other discussion on the boards before it is 'officially' announced.

If you like Twitter, I also tend to tweet these announcements so you can follow me there (see sig below).

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