Largest Prime

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:

• Source 1
• Source 2

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.