# Is (2^333)-1 a prime number? [closed]

How can I see if $2^{333}-1$ is a prime number?

Does this have to do with Mersenne prime numbers ($2^n-1$) ??

Thank you!

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 Ask Wolfram|Alpha! – Thomas Jan 31 at 21:13

## closed as too localized by Thomas, mikeazo♦Jan 31 at 21:42

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No, $2^{333} - 1$ cannot be a prime. It is easy to see this via the exponent, $333$, which has factorization $3\cdot3\cdot37$. From this we know, by a well-known identity, that

$$2^{333} - 1 = (2^{37} - 1)\cdot (1 + 2^{37} + 2^{37\cdot2} + 2^{37\cdot3} + 2^{37\cdot4} + 2^{37\cdot5} + 2^{37\cdot6} + 2^{37\cdot7} + 2^{37\cdot8}).$$

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 It's also divisible by $2^3 - 1 = 7$ by the same argument. i.e. It's not even close to prime. – Joe Z. Feb 1 at 20:38

No it's not prime. Its prime factorization (in ascending order) is

$7×73×223×1999×10657×169831×321679×1238761×26295457×36085879×199381087×319020217×616318177×698962539799×4096460559560875111$

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 Source:mathematica (and wolframalpha can do it too) – AFS Jan 31 at 21:20