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I recently came to know about the currently world's biggest prime number:

$$2^{74207281} − 1$$

This number has about $22.3$ million digits but how did the "Great Internet Mersenne Prime Search" found that particular number as on the date of asking this question. Was it some formula or algorithm they used to check for these massive prime numbers or something completely different?

Sources:

Numberphile
Wikipedia

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    $\begingroup$ It's not the biggest prime number, it's the biggest known prime number. There are bigger ones, we just don't know what they are yet. $\endgroup$ – Rahul Nov 6 '17 at 15:03
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Mersenne numbers are numbers of the form $2^n-1$. There exists a relatively fast test for their primality called the Lucas-Lehmer test, so checking their primality is less computationally expensive than checking that of "most" integers (ones without a nice form such as this).

However, the computation involved in this sort of primality testing is still massive, so it's usually done via distributed computation. This number was tested through the Great Internet Mersenne Prime Search.

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  • $\begingroup$ That's crazy how they found it! $\endgroup$ – Akshit Sinha Nov 6 '17 at 15:03
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    $\begingroup$ Well the search is aided by knowing to look only for exponents $n$ that are themselves prime. So it is far from a random search. $\endgroup$ – hardmath Nov 6 '17 at 15:28

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