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Hi Majortim,
The original proof can be found in a 1958 paper by Hans Riesel (Riesel, Hans (1956). "Några stora primtal". Elementa 39: 258–260). 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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Twitter: IainBethune
Proud member of team "Aggie The Pew". Go Aggie!
3073428256125*2^1290000-1 is Prime! | 
