Every year or so, the Great Internet Mersenne Prime Search announces a new largest known prime number. In 2001, the largest known prime had about 4 million digits. In 2008, we blew past the 10 million digit threshold. In 2018, a computer found a prime number (currently the largest known) with almost 25 million digits. Who knows what the next year will bring?
These primes are impressive, to be sure. I do not begrudge them that. I’ve even had my fun figuring out how to describe and write them. But amidst the fanfare each new largest known prime garners, there is a faithful friend who has always been there for us, and frankly has been a bit neglected: 2.
Not all mathematicians in history have considered 2 to be the smallest prime number. Some have held that 1 deserves that honor, but contemporary mathematicians are careful to define a prime number to be a positive whole number that has exactly two distinct divisors, itself and 1, excluding 1 from primality. With this definition, 2 is unlikely to be replaced as smallest known prime number any time soon.
The virtues of 2 are as great as 2 itself is small. It is the only even prime. In fact, it is the only prime with a commonly-used English word describing numbers it divides. (At least until I succeed at getting the word “threeven” into the lexicon. Come on. Threeven. You know you love it.) Two is also the only positive number whose square is its double. It is the only positive number n for which the sum of two powers of n can also be a power of n. (For example, 23+23=24, but there is no way to add two powers of 3 to get another power of 3.)*
And then there’s binary, also known as base two. Its simplicity is a balm in our complicated world. Every natural number written in binary begins with the digit 1. Every digit in binary is 1 or 0, yes or no, on or off. No shades of gray afflict binary. But binary is powerful as well. These zeroes and ones, operating as simple ons and offs, underlie the logic gates that create computer circuits and therefore every computer program we use.
But 2 is not all about comfort and power. In research-level algebra, 2 throws a wrench into things. Large swaths of these fields are devoted to looking at polynomials or functions that take inputs and produce outputs over what are known as finite fields. A finite field is a finite collection of numbers that allow the basic operations of arithmetic: addition, subtraction, multiplication, and division. If you are familiar with modular arithmetic, or clock arithmetic (the idea that 5 hours after 10:00 can also be called 3:00), finite fields are kind of like that, but turned up a notch. The number of elements in a finite field is always a power of a prime p. That p is called the “characteristic” of the field. Theorem after theorem involving finite fields requires the caveat “except for characteristic 2.” Why is 2 such an algebra spoiler? There are a few reasons—small numbers are weird, basically—but the upshot is that 2 keeps algebraists on their toes, and that is certainly a valuable service.
Two’s influence extends from its lowly position on the number line to the heady realms of the largest known primes. For about 65 of the last 68 years, the largest known prime number has had a particular form; each has been a Mersenne prime. That is not because there are a lot of Mersenne primes but because there is a fast-acting primality test that, when it finds a prime, always finds this kind of prime. Mersenne primes have the form 2p−1; each one is 1 less than a power of 2. The smallest known prime is inextricably linked to the largest.
*These sentences were edited after publication for clarity.