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I'm watching this, he says that David Slowinski discovered the biggest prime in 1984: $2^{132,049}$-1 and that it took 1 week on a Cray supercomputer: using some shortcut and that the absence of this shortcut would make the process last the age of the universe. What's this shortcut?

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Numbers of the form $2^n-1$ can be tested by the Lucas-Lehmer method, which is not available for numbers not of that form.

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Can you give me a brief description of this method? –  Vÿska Dec 6 '12 at 2:25
    
I have edited in a link to the description at Wikipedia, but you could just type $$\rm Lucas\ Lehmer$$ into the internet and take your pick of places to look. –  Gerry Myerson Dec 6 '12 at 2:29
    
I know I can, I just thought that it would be something deadly complicated and if this was true, I'd need a simplification. –  Vÿska Dec 6 '12 at 2:30
    
@Gustavo Take a look at it. It's not complicated at all. –  Rick Decker Dec 6 '12 at 2:34
    
Yep, I'm reading about it. Thanks for your answer. –  Vÿska Dec 6 '12 at 4:46

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