Prime Numbers

The loneliest numbers in math turn out to be the ones holding up everything else.

By Greg Goodwin · Published October 17, 2026 · Updated October 24, 2026

Cheat Sheet

The first primes: 2, 3, 5, 7, 11, 13, 17, 19, 23. The number 2 is the only even prime.
The number 1 is not prime: by definition a prime must be greater than 1.
Twin primes differ by 2, like 11 and 13. Whether infinitely many exist (the twin prime conjecture, posed by Alphonse de Polignac in 1846) is still unproven.
Mersenne primes have the form 2^p - 1. Fifty-two are known, though the numbering of the newest is provisional until all candidates below it are checked.
Leonhard Euler showed in 1772 that 2^31 - 1 (10 digits) is prime. The 1996 GIMPS find had 420,921 digits; the 2024 record has about 97 times as many.
Prime-hunting by the volunteer project GIMPS has produced every Mersenne prime found since November 1996.

The 60-Second Version

A prime number is a whole number greater than 1 that can be divided evenly only by itself and 1, like 2, 3, 5, 7, and 11. Every other whole number above 1 is a composite, and it can be broken down into primes in exactly one way, which is why primes are often called the multiplicative building blocks of the number system. Around 300 BCE, Euclid gave the first known proof that the primes never run out. Yet no formula reliably produces them, so mathematicians find big ones by searching. As of the latest published list, the largest known prime is 2^136,279,841 - 1, a number with 41,024,320 digits, found in October 2024 by Luke Durant through the GIMPS volunteer project using graphics processors. For centuries primes looked like pure curiosity. Then cryptographers discovered that large primes could be used to build nearly unbreakable codes, and now they quietly protect online communication.

The Long Version

What Makes a Number Prime

Take any whole number bigger than 1 and try to split it into smaller factors. Some numbers refuse: 7 can only be written as 7 times 1. Those are the primes. The rest, like 60, break apart, and keep breaking until only primes are left: 60 is 2 x 2 x 3 x 5. The fundamental theorem of arithmetic says this final list of prime factors is unique for every number, which is why primes are treated as atoms of arithmetic. The definition requires a number greater than 1, so 1 is excluded, and 2 is the only even prime because every other even number is divisible by 2.

Euclid and the Infinity of Primes

Primes were studied by the Greek mathematicians Euclid, who flourished around 300 BCE, and Eratosthenes of Cyrene (about 276 to 194 BCE). In his Elements, Euclid gave the first known proof that there are infinitely many primes. The idea is elegant: suppose you had a complete list of primes, multiply them all together, and add 1. The result leaves a remainder of 1 when divided by anything on your list, so it must be divisible by some prime that was not on the list. For example, 2 x 3 x 5 x 7 x 11 x 13 + 1 is 30,031, which is not prime itself but equals 59 x 509, and both of those primes are new. No finite list can ever be complete.

No Formula, So We Hunt

Over the centuries various formulas for generating primes have been proposed, but Britannica says all have been flawed. The most productive hunting ground has been Mersenne numbers, of the form 2^p - 1. Euler proved 2^31 - 1 prime in 1772, and Edouard Lucas proved 2^127 - 1 prime in 1876. Computers took over in 1952, when Raphael Robinson found five Mersenne primes in a single year. In 1996 the Great Internet Mersenne Prime Search (GIMPS) began recruiting volunteers' home computers, and its first discovery had 420,921 digits. Records kept climbing: 24,862,048 digits in 2018, then 41,024,320 digits in October 2024, found on an NVIDIA A100 graphics card. The list counts 52 Mersenne primes, though the newest numbering is provisional because not every candidate between the 2018 and 2024 finds has been ruled out.

Open Questions

Primes are infinite but irregular. Britannica names the prime number theorem and the Riemann zeta function as two famous results about how they are distributed. Twin primes, such as 17 and 19, are pairs that differ by 2, and mathematicians believe there are infinitely many but have not proved it. In 2013 Yitang Zhang proved that infinitely many prime pairs exist with a gap below 70 million, a bound that was later improved to 246. That is still far from a gap of 2. The Riemann zeta function is also tied to the famous Riemann hypothesis, which remains unsolved.

Primes at Work

For centuries number theory like this was thought to have no practical use. Then cryptographers discovered how large primes could be used to make nearly unbreakable codes, as Britannica describes in its entry on two-key cryptography. The core idea is an asymmetry: multiplying two big primes is easy, while recovering them from the product is extremely hard. That lopsided difficulty is what lets strangers exchange secrets across open networks, which is why giant primes matter far beyond record-chasing.

Why People Care

Primes hide some of mathematics' oldest unsolved puzzles and give volunteers with ordinary computers a shot at a world record. They also underpin modern cryptography, so a topic that began with Greek geometry now helps secure the messages and payments of everyday life.

Glossary

Prime number: A whole number greater than 1 that is divisible only by itself and 1.
Composite number: A whole number greater than 1 that has at least one divisor besides itself and 1, so it is a product of smaller primes.
Fundamental theorem of arithmetic: Every whole number greater than 1 can be written as a product of primes in exactly one way, apart from order.
Mersenne prime: A prime of the form 2^p - 1, named for the French scholar Marin Mersenne.
GIMPS: The Great Internet Mersenne Prime Search, a volunteer computing project that has hunted record primes since 1996.

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