This Is the Largest Known Prime Number Yet

Meet the new largest known prime number. It starts with a 4, continues on for 23 million digits, then ends with a 1. As is true with all prime numbers, it can only be evenly divided by one and itself.

Prime numbers are essential to modern life, used in everything from securely encrypting banking information to the random number generators used by visual effects specialists for the latest movies. And while finding larger prime numbers doesn't necessarily mean stronger encryption (that's a common misconception) human curiosity drives the continual quest to find ever-larger primes.

"Each new prime is an extension of the bounds of human mathematical knowledge," Hartree Centre researcher Iain Bethune, who is part the prime number hunting project PrimeGrid, which was not involved in the new find, writes in an email to Smithsonian.com.

The newest prime number is generated by multiplying two by itself 77,232,917 times, then subtracting one. In mathematical terms that is: 2^77,232,917 - 1. This format of calculation means the new prime is considered a Mersenne prime. Named after the French theologian and mathematician Marin Mersenne, these types of primes are always calculated as a power of two minus one. This pattern creates a countable (although still enormous) list of candidate Mersenne prime numbers.

The number—which can be written in shorthand as M77232917—is nearly one million digits longer than the last confirmed prime discovered in 2016. While it’s the fiftieth Mersenne prime discovered, not all candidates between the last two primes have yet been checked so another could be lurking between them. But that would be surprising, says Chris Caldwell, a mathematician who tracks the discovery of large prime numbers. According to Caldwell, the gap between Mersenne primes is usually much larger.

When M77232917 is written out as all 23,249,425 digits, the number contains every digit from zero through nine roughly 2.3 million times each. And like all prime numbers, it appears to be random, although some researchers suggest that faint patterns shape the distribution of prime numbers.

These faint patterns are enough to help narrow the search for new prime numbers. This helps researchers predict how many primes will exist within a range of numbers, explains Robert Lemke Oliver, a mathematician at Tuffts Univerisity. "It happens that among numbers with 1000 digits, about one in every 2500 will be prime," he writes in an email to Smithsonian.com.

Discovering the new prime was a group effort. A computer owned by Jonathan Pace, an electrical engineer living in Tennessee, identified the number using specialized Great Internet Mersenne Prime Search (GIMPS) software. Developed by George Woltman, the software tests candidate numbers as part of a search coordinated by PrimeNet system software, which was written by Scott Kurowski and maintained by Aaron Blosser. After its discovery, M77232917 was verified as a prime number by Blosser and three other people—David Stanfill, Andreas Höglund, and Ernst Mayer—each using different software and computer setups.

"What's special about this prime isn't that it's prime, it's that we actually know it's prime," writes Lemke Oliver. Determining if a number is a prime is conceptually simple. All you need to do is divide it by all primes smaller than itself. If no other primes can divide it evenly, it must be a new prime number. In practice, however, this brute-force approach is time-consuming for extremely large numbers, even with modern computers capable of exquisitely quick calculations. Instead, algorithms take advantage of a number theory trick called the Lucas-Lehmer test that only works for Mersenne primes to speeds up the process.

Even so, it is still computationally exhausting to test prime number candidates. Pace’s computer took six days of dedicated time to discover M77232917; the verifications took an additional 291 computing hours. The discovery is a first for Pace, who has been running software to hunt for big prime numbers for the past 14 years.

Finding new prime numbers is a hot topic. GIMPS offers research awards for the discovery of new Mersenne prize numbers (Pace won $3,000 for his recent discovery), while the Electronic Frontier Foundation has a series of open challenges for the first to discover primes of ever-increasing magnitudes. GIMPS estimates it will take 15 years of calculations to reach the next milestone, finding a prime number that is at least 100 million digits long.

The motivation of the prize, set up in the 1990s, is quaint in a modern context, says Seth Schoen of the Electronic Frontier Foundation. "The awards are meant to show how the Internet is useful—to let people who may never have met work together on a large scale to accomplish things," he writes in an email. 

And that collaboration is key for finding these big primes. "A single person with a shovel might find a large gem, but it is very unlikely," writes Caldwell. "But if you can organize 100,000 people with shovels, coordinate where and how they dig, the chance of the group finding a gem is far far higher." Software like PrimeNet hands out the shovels and coordinates digging sites, while GIMP does the digging.

Welcome to the list of primes, M77232917, and enjoy your time as the largest prime number while you can. Just like death and taxes, one thing is certain: one day, a new largest prime number will be discovered.

Mika McKinnon | READ MORE