# Writing digits of a large prime in a highway

The digits in the above picture of a highway are the first digits of the largest prime number discovered almost two weeks ago on December 26, 2017. The number has 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 digits would also fill an entire shelf of books for a total of 9,000 pages!

Though the previous largest prime number is plenty large, it digits would only cover a stretch of highway about 70.5 miles long! So the new discovery would cover a further distance of about 2.5 miles.

The brand new largest prime is the number 2 raised to 77,232,917 less 1. Prime numbers that are generated in this fashion are called Mersenne primes, in honor of the French monk Marin Mersenne, who studied these numbers more than 350 years ago. The previous largest known prime number is also a Mersenne prime.

Indeed, 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). In fact, anyone who has a computer and has Internet access can do it. It does not take special mathematical expertise. It takes patience. The discoverer of this latest record, Jonathan Pace, is a GIMPS volunteer for over 14 years. The GIMPS site would have all the information to get started for anyone who wants to join the search.

Why the fascination with prime numbers? Some people think of finding the next largest known prime number as Climbing Math Everest. The best thing about this Everest is that there is another Everest to conquer after one is successfully scaled. Euclid proved over 2,000 years ago that there are infinitely many such Everest. 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.

Now we know that the current world record of the largest prime number is $2^{77,232,917}-1$, which has 23,249,425 decimal digits. This Mount Everest was scaled in December 2017. Volunteers at GIMPS are hard at work in trying to find the next Mount Math Everest.

The theoretical aspects of prime numbers are even taller Math Everest than the computational ones. Many problems involving prime numbers are easy to state but hard to prove. Some have withstood attempts at solving for centuries (twin prime conjecture and Goldbach conjecture come to mind).

Prime numbers have real world practical implications too. Online commerce is made possible by the use of prime numbers in keeping information secure. Though the large prime numbers used in cryptography are much smaller in comparison to the largest prime number discovered recently. Who knows. It could be possible that prime numbers with millions of digits could find practical use in the future.

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$\copyright$ 2018 – Dan Ma