How Large is the Largest Prime Number?

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For almost 2,000 years, mathematicians have been searching for prime numbers. Even in the time of the ancient Greek mathematician Euclid, humans were fascinated by finding primes. Many have been found, but since prime numbers are infinite, the search goes on. Of the primes found to date, what is the largest? And how are such huge primes discovered?

Well, before discussing the largest prime, let's understand what a prime number is. A prime number is that which cannot be divided by any number other than 1 and itself. The simple meaning of the statement is that it is divisible only by 1 and itself. For example, 7 is a prime number because it can't be divided into parts by any number besides 1 and 7.

Every integer greater than 1 is either a prime or a product of primes. For instance, 18 is not a prime, but all of its divisors are primes: 2×3×3=18. And this is true for all numbers.

The currently largest known prime number is . While this is easy to write, finding this number is no easy task—it's nearly impossible without computers. To put into perspective just how large this number is, let's walk through the process of finding it.

Take 2 and multiply it by itself 82,589,933 times. For instance,  and 2^3=2×2×2=8. To find the largest prime, you would multiply 2 by itself some 82 million times, then subtract 1. The result has 24,862,048 digits—1.5 million digits more than the previous largest prime. Obviously, this isn't something you can do with pen and paper or even a typical calculator. 

This massive prime has a name: M82589933. It's a member of the special class of primes known as Mersenne primes. A Mersenne prime is a number of the form 2n − 1, where n is a positive integer, and the result is a prime number. For instance, 31 is a Mersenne prime because 31 = 2 5 − 1 = 2 5 −1 (where 2 5 = 32 2 5 = 32). But not all numbers of the form 2n − 1 are

−1 are prime.

For decades, mathematicians have been testing Mersenne primes to confirm whether they are, indeed prime. Finding primes is challenging, but working with Mersenne primes makes the task a bit more manageable. Since the start of 2018, a new Mersenne prime was discovered almost every year thanks to the Great Internet Mersenne Prime Search (GIMPS). The largest prime was also uncovered by a group of hobbyists using GIMPS. Later, three others confirmed the number independently using other computers and software packages. No more Mersenne primes have been discovered since then. When the next one will be found is anybody's guess.

What is the Point of Finding Prime Numbers?

Prime numbers have great applications. They are at the basis of cryptography, ensuring safe communication over the net. Primes play an important role in the design of data structures and software and are vital in number theory in mathematics. Beyond mathematics, primes find applications in physics, chemistry, and biology as well.


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