A new prime has been found (2018)

The GIMPS project has discovered the largest prime to date... it's a Mersenne prime, which means it is of the form 2^n − 1 for some whole number n. ☺

In this case, it is 2^77,232,917 − 1. This number has 23,249,425 digits, and it's nearly one million digits longer than the previous Mersenne prime found.

Read about the discovery here.

The work to find these really large primes is done by running specialized software on many regular computers -- in fact, anyone with a modern computer can help in this project!
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