Imagine a number made up of a vast string of ones: 1111111…111. Specifically, 136,279,841 ones in a row. If we stacked up that many sheets of paper, the resulting tower would stretch into the stratosphere.

If we write this number in a computer in binary form (using only ones and zeroes), it would fill up only about 16 megabytes, no more than a short video clip. Converting to the more familiar way of writing numbers in decimal, this number – it starts out 8,816,943,275… and ends …076,706,219,486,871,551 – would have more than 41 million digits. It would fill 20,000 pages in a book.

Another way to write this number is 2^136,279,841 – 1. There are a few special things about it.

First, it’s a prime number (meaning it is only divisible by itself and one). Second, it’s what is called a Mersenne prime (we’ll get to what that means). And third, it is to date the largest prime number ever discovered in a mathematical quest with a history going back more than 2,000 years.

The discovery

The discovery that this number (known as M136279841 for short) is a prime was made on October 12 by Luke Durant, a 36-year-old researcher from San Jose, California. Durant is one of thousands of people working as part of a long-running volunteer prime-hunting effort called the Great Internet Mersenne Prime Search, or GIMPS.

A prime number that is one less than some power of two (or what mathematicians write as 2 ^p – 1) is called a Mersenne prime, after the French monk Marin Mersenne, who investigated them more than 350 years ago. The first few Mersenne primes are 3, 7, 31 and 127.

Durant made his discovery through a combination of mathematical algorithms, practical engineering, and massive computational power. Where large primes have previously been found using traditional computer processors (CPUs), this discovery is the first to use a different kind of processor called a GPU.

GPUs were originally designed to speed up the rendering of graphics and video, and more recently have been repurposed to mine cryptocurrency and to power AI.
Durant, a former employee of leading GPU maker NVIDIA, used powerful GPUs in the cloud to create a kind of “cloud supercomputer” spanning 17 countries. The lucky GPU was an NVIDIA A100 processor located in Dublin, Ireland.

Primes and perfect numbers

Beyond the thrill of discovery, this advance continues a storyline that goes back millennia. One reason mathematicians are fascinated by Mersenne primes is that they are linked to so-called “perfect” numbers.

A number is perfect if, when you add together all the numbers that properly divide it, they add up to the number itself. For example, six is a perfect number because 6 = 2 × 3 = 1 + 2 + 3. Likewise, 28 = 4 × 7 = 1 + 2 + 4 + 7 + 14.

For every Mersenne prime, there is also an even perfect number. (In one of the oldest unfinished problems in mathematics, it is not known whether there are any odd perfect numbers.)

Perfect numbers have fascinated humans throughout history. For example, the early Hebrews as well as Saint Augustine considered six to be a truly perfect number, as God fashioned the Earth in precisely six days (resting on the seventh).

Practical primes

The study of prime numbers is not just a historical curiosity. Number theory is also essential to modern cryptography. For example, the security of many websites relies upon the inherent difficulty in finding the prime factors of large numbers.

The numbers used in so-called public-key cryptography (of the kind that secures most online activity, for example) are generally only a few hundred decimal digits, which is tiny compared with M136279841.

Nevertheless, the benefits of basic research in number theory – studying the distribution of prime numbers, developing algorithms for testing whether numbers are prime, and finding factors of composite numbers – often have downstream implications in helping to maintain privacy and security in our digital communication.

An endless search

Mersenne primes are rare indeed: the new record is more than 16 million digits larger than the previous one, and is only the 52nd ever discovered.

We know there are infinitely many prime numbers. This was proven by the Greek mathematician Euclid more than 2,000 years ago: if there were only a finite number of primes, we could multiply them all together and add one. The result would not be divisible by any of the primes we have already found, so there must always be at least one more out there.

But we don’t know whether there are infinitely many Mersenne primes – though it has been conjectured that there are. Unfortunately, they are too scarce for our techniques to detect.

For now, the new prime serves as a milestone in human curiosity and a reminder that even in an age dominated by technology, some of the deeper, tantalising secrets in the mathematical universe remain out of reach. The challenge remains, inviting mathematicians and enthusiasts alike to find the hidden patterns in the infinite tapestry of numbers.

And so the (mathematical) search for perfection will continue.

John Voight, Professor of Mathematics , University of Sydney. Republished from The Conversation.