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Do Mersenne numbers turn out to be interesting in other fields of mathematics besides Number Theory?

In other words, are the primes occuring in the sequence $$M_n=2^n-1$$ or the recursive realtion $$M(n+1)=2\times M(n)+1$$ useful in other fields of mathematics to solve problems or appear in the definition of other important functions?

I apologize in advance for any mistakes in my English, I hope that the translator did a good job :).

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They're great for making really bad RSA keys. – Alexander Gruber Mar 22 '13 at 21:09
see this Wikipedia page... – Coffee_Table Mar 22 '13 at 21:28
What do you mean? Using two mersenne numbers in the RSA algorithm generates weak keys? – MphLee Mar 22 '13 at 21:35
@Coffee_Table :thanks but the section "Mersenne numbers in nature and elsewhere" is not very rich, for that I asked here. – MphLee Mar 22 '13 at 21:54
A generalization is the "repunits": Mersenne numbers are numbers of the form "111...1" in base 2; the analoguous exists for other bases. Let b be another base then ${ b^n - 1 \over b - 1}$ gives numbers, which are repunits in the digitrepresentation of that number system. So search for keywords "repunits" and also "cyclotomic polynomials" to find possibly more interesting stuff ... – Gottfried Helms Mar 23 '13 at 9:07

Quoting from Wikipedia, "The Mersenne twister is a pseudo random number generator.... It provides for fast generation of very high-quality pseudorandom numbers, having been designed specifically to rectify many of the flaws found in older algorithms. Its name derives from the fact that period length is chosen to be a Mersenne prime."

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