# How long is 2 to 77,232,917 minus 1

The largest known prime seems to be updated about once every two years. On December 26, 2017, the newest largest known prime number was discovered. The prior discovery was in January 7, 2016 (it was found by a machine on September 17, 2015 but no human took note of it until January 7, 2016).

The first digits of 2 to 77,232,917 minus 1

Like the previous largest prime, this freshly found largest prime number is of the form $2^p-1$. That is, it is a power of 2 minus 1. In particular, the $p$ for this new largest prime is 77,232,917. So the number is obtained by multiplying 2 by itself 77,232,917 times and subtracting 1.

This number consists of over 23 million decimal digits (23,249,425 to be precise). If writing 5 digits per inch, all the digits of this prime number would cover a stretch of highway over 73 miles in length! That’s only 5 miles under the distance of three marathons.

The previous largest prime number that was discovered in January 2016 consists of only about 22 million digits (22,338,618 to be precise). These would cover about 70.5 miles. So the newly discovered largest prime number would stretch out a further distance of about 2.5 miles!

Here’s the first 120 and the last 120 digits of this new largest prime number.

4673331833592310999883355855611155212513
2110281771449579858233859356792348052117
7207484311099740208849621368090038049317 …

(the middle 23,249,185 digits omitted)

… 2853760045187860554022233766729256792821
3196546734339594539737047636927989462799
9939614659217371136582730618069762179071

Any prime number of the form $2^p-1$ is called a Mersenne prime, in honor of the French monk Marin Mersenne, who studied these numbers more than 350 years ago. The latest largest prime is a Mersenne prime. So is the one before that. In fact, most of the record largest primes are of this type. There is a large worldwide community of volunteers who devote their free time in hunting for Mersenne primes, called Great Internet Mersenne Prime Search (GIMPS for short).

The prime number $2^{77232917}-1$ is the 50th known Mersenne prime. Its discoverer, Jonathan Pace, is a GIMPS volunteer for over 14 years. The recent discovery is exciting news. The discovery of such a large prime is akin to the scaling of a new and higher Mount Everest. The even more exiting news is that there are plenty more Mount Everest waiting to be discovered and scaled. Euclid proved around 2000 years ago that there are infinitely many prime numbers. Theoretically we know there is no such thing as the largest prime. When we speak of the largest prime, it is only the largest prime number verified by the computing resource that is currently available.

For more information about the latest new largest prime, see the website for GIMPS or Google the Internet. The news of this new discovery is reported in this piece from npr.org. A piece in a companion blog has a basic discussion on Mersenne primes.

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