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The 64 bit machine's biggest number can be 2^63 (9,223,372,036,854,775,807). So what kind of a monstrous machine can generate 100-400 digits prime number? I probably don't understand something, so that's why I am asking.

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yes, the 64 bit machine's biggest number is $2^{63}$, but with this machine you can describe bigger number than that with some method like this $a=\sum_{i=0}^{n}a_{i}B^{i},0\le a_{i} <B$, where $B$ can be the biggest number in your machine, then you can test the primality of it –  Nax Oct 13 '13 at 23:58
    
On the most 64-bit machines the biggest number representable is actually 2^64-1 ( = 18446744073709551615). Using formula above (see comment from Alex) it is possible to represent larger numbers. This formula works equally well for smaller processors. In fact, even some 8-bit processors have been used to calculate RSA using such long prime numbers. –  user4982 Oct 14 '13 at 14:59
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Your human hands only have 10 fingers. What kind of monstrous creature can do arithmetic on 2-digit, 3-digit, or even larger numbers in base 10? –  Nemo Oct 14 '13 at 18:04
    
@Nemo Muahahahaha. –  evening Oct 14 '13 at 18:07

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You can simply combine mathematical operations on 64 bit numbers to create mathematical operations on bigger numbers. Just like a computer can combine bit operations to create operations on bytes and - yes - 64 bit values. In the end it is all binary arithmetic, calculations in current computers are performed on values of just 0 and 1.

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For an example, check out the implementation of the Java BigInteger class; internally it consists of an array of longs, which can easily be seen in any Java debugger. –  owlstead Oct 14 '13 at 17:36

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