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3
1
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This may be subjective, but does anyone have any insight into why this is the case? This struck me while considering that it's also the eigth Mersenne prime (2^31-1=2147483647).
UPDATE: It's been pointed out that the relationship doesn't necessarily hold for larger storage classes, e.g., 2^63 - 1 is not prime. |
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5
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Why is $3$ prime? I don't really know that there are meaningful answers to these kinds of questions. The best I can think of is some reasons it is not obviously composite, e.g. since $5$ is prime $2^5 - 1 = 31$ is not obviously composite (and it turns out to be prime) hence $2^{31} - 1$ is not obviously composite. This is two applications of the "lemma" that if $p$ is prime then $2^p - 1$ is not obviously composite. Note that any prime factor of $2^p - 1$ has to be congruent to $1 \bmod p$ by Fermat's little theorem, so it is "easier" for such numbers to be prime. |
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0
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The Largest 64 bit prime is 18446744073709551557, according to http://msdn.microsoft.com/en-us/library/ee621251.aspx. See also http://primes.utm.edu/lists/2small/0bit.html |
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