Why do we keep looking for prime numbers beyond 22 million digits?

In December 2018 the last of the new Mersenne primes, a milestone that once again gave prominence to these special numbers that always have the same format (2p-1) and that each time they have a greater number of digits. In fact the number found, 274.207.281-1, has more than 22 million digits.

The discovery seems more anecdotal than anything else, and although these numbers are almost an obsession for mathematicians, the reality is that finding new prime numbers makes sense in various practical and theoretical fields.

Evaluating the raw power of processors

The process for discovering new Mersenne primes is especially demanding. Programs like Prime95 are used to evaluate gross performance of modern processors and allow us to know if these chips are capable of supporting very high workloads for long periods of time.

This is an especially popular tool among overclocking fans, which can determine whether by forcing the CPU or other components to perform stably even under these high workloads.

In fact, this type of computational process helped find a bug in the Skylake processors from Intel – Working with Prime95 found that these processors could hang or cause unpredictable system behavior. Intel recognized the problem and has fixed it through a BIOS update that has been distributed in collaboration with its partners in the field of motherboards.

A virtual supercomputer that surpasses all current ones

This Mersenne search for prime numbers has also become a demonstration of just how much the distributed computing. He GIMPS project (Great Internet Mersenne Prime Search) makes use of the Prime95 software precisely so that the workload of searching for new prime numbers is distributed among a huge number of computers all over the world.

Gimps2

Anyone can join the project, and in fact by running the Prime95 program We are asked if we want to join this initiative or just pass a “stress test” with which to precisely assess the stability of our team. It is one of the best-known distributed computing projects in the world, and although originally Prime95 focused on the CPU, a GPGPU version has been in the works for a long time that precisely takes advantage of the enormous raw power of dedicated graphics cards.

This project has been responsible for the discovery of several new Mersenne primes, and currently has a throughput (media of the last 30 days) of 946.781 TFLOPs.

The figure is really remarkable and would make this “distributed computer” the most powerful in the world, above all the supercomputers on the Top500 list. Number one of that list is Summit, in the United States, with a peak power of 200,794 TFLOPS.

On December 21, 2018 it was discovered precisely the largest known prime number, 282,589,933-1 — he’s the 51st Mersenne cousin in all of history — which has no less than 28,862,048 digits.

It was discovered by one of the volunteers from this global effort -at present, more than two million machines are registered in that singular project- that precisely used this algorithm to test its systems in scenarios of computational stress. A few months ago it was discovered another one of these prime numbers, but the latter is even bigger and therefore more striking.

Today we use it to encrypt communications, tomorrow … who knows

How they explained and Ars Technica, one of the practical applications today of these prime numbers is the RSA encryption: the person who wants to receive a message protected with this algorithm will publish the product of two large prime numbers as his “public key”, something that makes it very difficult to decrypt with brute force.

Haiku Proof

Source: xkcd. Explanation of the comic strip, here.

A person potentially interested in those encrypted messages would have to try to guess the proper prime numbers with which the public key was generated, something that can be very laborious.

Also, the larger the prime numbers used, the more potential combinations there are to create the public key and more difficult is to find that public key.

The secret of success here is because there can be no confusion in factoring: whereas if any two numbers were used there could be other integers with which to obtain the key. When using prime numbers those possibilities are finite, and that makes the discovery of new prime numbers especially interesting.

Numbers

The reasons to continue advancing in that search, as argued Dr. Chris Caldwell, professor of mathematics at the University of Tennessee are very diverse, and not necessarily “practical”.

Some do it out of tradition, out of a fondness for collecting strange objects or landmarks, and even for glory (and money, find a cousin of Mersenne you pay) of having your name associated with the discovery of a new Mersenne prime number.

However, it is one of those processes that we don’t even know if it will be more useful in the future, something that has happened with other discoveries in the past that were later applied to developments that were not even in the mind of their creators when the original discoveries occurred. So, it seems like it wouldn’t be a bad idea for you to look for those new prime numbers too, don’t you think?

Update (December 2018)– We’ve added text to discuss the latest Mersenne primes discovered in 2018, as well as introducing some interesting links throughout the entire text to extend this reading.

Update (January 2020): we have slightly updated the text and added more interesting links to complete it.

In Xataka | “Encryption does not mean that communication is completely secure” Interview with Phil Zimmermann


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