The news of the discovery of a new, largest known prime broke about a week ago but I've only gotten around to writing about it here. It was of course a Mersenne prime discovered via GIMPS, the Great Internet Mersenne Prime Search.
The number containing \(22,338,618\) digits is \(2^{74,207,281} - 1\) where \(74,207,281\) itself must be prime of course. It is the 49th known Mersenne prime defined as a prime expressible in the form \(2^p - 1\) where \(p\) is prime. The first Mersenne primes are 3, 7, 31, and 127 corresponding to \(p\) values of 2, 3, 5, and 7 respectively.
Note that p being prime is not sufficient to ensure that 2^p - 1 will be prime. As a counter example take \(p=11\). The resulting number \(2^{11} - 1 = 2047 = 23 \times 89\) is not prime. Here are links to some more interesting information about Mersenne primes:
on 27th of October 2024
Update: \(2^{136,279,841} - 1\) has \(41,024,320\) digits and is prime! Read all about the new largest prime number ever found: Stand-up MathsYouTube video.
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