The largest prime number ever discovered has been found by an amateur mathematician using Great Internet Mersenne Prime Search (GIMPS).
Prime numbers, as you likely learned in school, are numbers that can only be divided by one and themselves. There are an infinite number of them, with all numbers greater than 1 being either a prime number or a composite of primes. But having discovered a whole lot of them, new primes are now notoriously difficult to find.
Mathematicians are not totally ignorant of how to find them, nor forced to go through each individual number and check. Mersenne primes are an example of a prime with a nice pattern to them, as they can be expressed as 2-1, or 2 to the power of a prime, minus 1. Small ones include 31 (2-1) and 127 (2-1), but they scale pretty quickly - and recent ones have generally been found with the help of computer power.
The Great Internet Mersenne Prime Search (GIMPS) looks for these Mersenne primes, allowing anyone to download software and help search for them, with a $3,000 reward for anyone who comes across one. The last 18 Mersenne primes have been found by GIMPS, and now 36-year-old researcher and former NVIDIA employee Luke Durant has found the largest Mersenne prime so far by developing infrastructure that can run GIMPS across many GPU servers.
"Physicists keep talking about, you know, information in the universe being an important first principle, so let me go try to find a new unique piece of big information and see if that helps guide my thinking about large numbers," Durant told Numberphile, explaining his motivation to try to find it.
"It was it was it actually pretty exciting to me to you know reach a scale of genuinely a global supercomputer put together in my office and [it] found a unique result. It's pretty fun," he added.
Proving numbers are prime gets really interesting. With smaller primes, say 11, it's an easy task. Simply divide it by all the smaller integers (1-10) and see if you are left with any integers. If it is only divisible by 1 and itself, it's a prime, as it can't be made from two smaller numbers. That's what a prime is. But for larger numbers, say 15,678,547,356,947 for example, you can see how it would be more time-consuming. Fortunately, mathematicians have some pretty neat tricks to test if a number is a prime or not, without resorting to working out if 15,678,547,356,947 is divisible by 3,187.
One method, described by Numberphile below, involves calling "witness" numbers to test if the number is prime. This gets complicated, with some numbers being better witnesses than others.
To test this prime number, GIMPS first performs a Fermat Primality Test, which can tell you whether the number is likely to be a prime. The new prime passed, but unfortunately, there are a small fraction of numbers known as "Carmichael numbers" which give false positives for prime numbers, meaning you can't be fully sure it is a prime using this method.
They then used the more definitive Lucas-Lehmer primality test for determining if Mersenne numbers are primes, finding again that the candidate is a prime. The team opted for October 12 as the date of discovery, the date that the Lucas-Lehmer test was run.
Credit for the discovery goes to Durant, Mihai Preda, and George Woltman for designing the software, and Aaron Blosser for maintaining the server. The new prime, as well as being the largest known prime number, adds to GIMPS' dominance in finding Mersenne primes, and becomes only the 52nd known Mersenne prime since they were first studied over 350 years ago.