What is the largest known prime number?
Prime numbers have been investigated for more than 2,000 years, since at least the era of the ancient Greek mathematician Euclid. There are infinitely many, but what is the largest known prime number?
Prime numbers are those that can be evenly divided only by 1 and themselves, such as 3 and 7. They are key building blocks in math; per the fundamental theorem of arithmetic, every number greater than 1 is either a prime number or a multiple of a prime number, according to the University of Houston.
"Prime numbers are the 'atoms' of number theory," Thomas Kecker, a mathematician at the University of Portsmouth in England, told Live Science.
A major difference between real atoms and prime numbers is that the number of different types of stable atoms is finite. In contrast, "it is known at least since the times of Euclid in ancient Greece that there is an infinitude of prime numbers," Kecker said. "Finding larger and larger prime numbers therefore became a quest for many mathematicians."
Related: How many atoms are in the observable universe?
Currently, the largest known prime number is 2^(82,589,933) - 1. To calculate this number, multiply 2 by itself 82,589,933 times, and then subtract 1. The result, also known as M82589933, possesses a whopping 24,862,048 digits, more than 1.5 million digits more than the previous record holder, according to the University of Nebraska-Lincoln.
M82589933 is a Mersenne prime, a kind of number named after the French monk Marin Mersenne, who investigated these numbers more than 350 years ago. To calculate a Mersenne prime, 2 is multiplied by itself a number of times, and then 1 is subtracted, according to the Great Internet Mersenne Prime Search (GIMPS).
GIMPS is a distributed computing project in which groups of volunteers run software in the background on their computers to collectively solve problems — in this case, finding Mersenne primes. Founded in 1996, GIMPS is the longest continuously running distributed computing project, according to the project website.
"This distributed computing approach to finding the largest known prime number has been very successful — the GIMPS group has found 17 Mersenne primes," Curtis Cooper, a mathematician retired from the University of Central Missouri, told Live Science. "Most of these were the largest known prime numbers at the time of their discovery."
Cooper and his colleagues have discovered four Mersenne primes, all of which were the largest known primes when they were found. M82589933 was discovered on Dec. 7, 2018, by a computer volunteered by Patrick Laroche, an IT professional living in Ocala, Florida, after 12 days of nonstop computing, according to GIMPS. Currently, GIMPS runs on more than 2.6 million CPUs performing about 4 million billion calculations per second.
"For a large whole number — say, with a few thousand digits — it becomes more and more time-consuming to check whether or not that number is prime," Kecker said. "Even with the most advanced algorithms and latest supercomputers to run them on, testing whether or not a number is prime could easily exceed a human lifespan."
However, over the years, mathematicians have discovered strategies for finding out if Mersenne numbers are prime, and these methods are far quicker than the techniques used for other kinds of prime numbers. Until 2018, GIMPS discovered a new Mersenne prime about every other year. "No new one has been found since," Kecker said. "It is almost like waiting for a volcanic eruption after a long period of inactivity — although one expects the next one to happen any time, one never knows when it strikes again, if it ever strikes again."
