What Are Prime Numbers, and Why Do They Matter?

So, what are prime numbers, anyway? And how did prime numbers get to be so important in the modern world? As Wolfram MathWorld explains, a prime number — also known simply as a prime — is a positive number greater than 1 that can only be divided by one and itself. It needs to be divisible by two numbers. With that definition of prime numbers in mind, the number 1 isn't a prime number.

A good way to remember it is to know that a prime number cannot be divided by any other positive natural numbers without leaving either a remainder, decimal or fraction. Take the example of the prime number 13. It only has two divisors: 1 and 13. 13 ÷ 6 = 2 with a remainder of 1. Dividing a prime number by any other natural number results in leftover numbers.

Throughout history, mathematicians have grappled with the concept of what truly defines a prime number. Central to this debate was the status of the number 1. In the 19th century, there was a debate over whether 1 is a prime number or not.

People once believed 1 to be prime. The foundation of this belief rested on the idea that a prime number is defined by having only two positive integer divisors: one and itself. Hence, the only integer that posed a challenge in categorization was 1, because, by this basic definition, it met the criteria.

However, as mathematics evolved, there was a shift in this perspective. To make number theories and their resultant theorems more consistent and coherent, mathematicians revisited the criteria for a number to be identified as prime. The concept of prime numbers needed a distinction between prime and composite numbers.

By the definition that a prime number has exactly two distinct positive divisors, the number 1 didn't fit since it only has one distinct positive divisor: 1. Therefore, the categorization changed, no longer considering 1 prime.

This shift ensured that every positive integer greater than 1 is classified as either prime or composite. It helped to provide clarity in mathematical theories and theorems, eliminating potential ambiguities. While the debate has largely been settled with the consensus that 1 is not a prime number, the historical debate underscores the evolving nature of mathematical definitions and the constant quest for precision in the discipline.

"The only even prime number is 2," says Debi Mink, a retired associate professor of education at Indiana University Southeast, whose expertise includes teaching elementary mathematics. "All the other prime numbers are odd numbers." This is because they have more than two factors. So, let's take a look at that.

All even numbers are composite numbers. 2 is the only even prime number because it doesn't have more than two factors — its only factors are 1 and the number 2 itself. For a number to be classified as a prime number, it should have exactly two factors. Since 2 has exactly two factors, 1 and the number itself, 2, it is a prime number.

Numbers like 2, 3, 5, 7, 11, 13 and 17, are all considered prime numbers because they have exactly two factors, 1 and the number itself. Numbers like 4, 6, 8, 9, 10 and 12 are not prime numbers because they have more than two factors.

Composite numbers are the opposite of prime numbers. They can be divided by other numbers besides 1 and themselves.

Mark Zegarelli, author of numerous books on math in the popular "For Dummies" series who also teaches test prep courses, offers an illustration involving coins that he uses with some of his students to explain the difference between prime numbers and composite numbers.

"Think about the number 6," says Zegarelli, citing a composite number. "Imagine that you have six coins. You could form them into a rectangle, with two rows of three coins. You can do that with eight, too, by putting four coins into two rows. With the number 12, you could make it into more than one type of rectangle — you could have two rows of six coins, or three times four."

"But if you take the number 5, no matter how you try, you can't put it into a rectangle," Zegarelli notes. "The best you can do it is string it into a line, a single row of five coins. So, you could call 5 a non-rectangular number. But the easier way to say that is to call it a prime number."

There are plenty of other prime numbers — 2, 3, 7 and 11 also are on the list, and it keeps rolling from there. The Greek mathematician Euclid, back in circa 300 B.C.E., devised a Proof of the Infinitude of Primes, which may have been the first mathematical proof showing that there is an infinite number of prime numbers. (In ancient Greece, where the modern concept of infinity wasn't quite understood, Euclid described the quantity of prime numbers simply as "more than any assigned multitude of prime numbers.")

Another way of understanding prime numbers and composites is to think of them as the product of factors, Zegarelli says. "2 times 3 equals 6, so 2 and 3 are factors of 6. So, there are two ways to make six — 1 times 6, and 2 times 3. I like to think of them as factor pairs. So, with a composite number, you have multiple factor pairs, while with a prime number, you have only one factor pair, one times the number itself."

Proving that the list of prime numbers is infinite isn't that tough, Zegarelli says. "Imagine that there is a last, biggest prime number. We're going to call it P. So then I'll take all the prime numbers up to P and multiply them all together. If I do that and add one to the product, that number has to be a prime."

If a number is a composite number, in contrast, it's always divisible by some quantity of lower prime numbers. "A composite could be divisible by other composites as well, but eventually, you can decompose it down to a set of prime numbers." (An example: the number 48 has exactly two factors, 6 and 8, but you can break it down further into more than only two factors: 2 times 3 times 2 times 2 times 2.)

The Sieve of Eratosthenes is a method, introduced by Greek mathematician Eratosthenes in the third century B.C.E., used to find the prime numbers and composite numbers among a group of numbers.

The Sieve of Eratosthenes is based on the idea that the multiples of a prime number are not prime themselves. So, when searching for prime numbers, all the multiples of each prime number can be crossed out. This eliminates many numbers that would otherwise have been tried for no reason, so the Sieve of Eratosthenes can save a lot of time.

There are only 25 prime numbers between the numbers 1 and 100:

So why have prime numbers held such a fascination among mathematicians for thousands of years? As Zegarelli explains, a lot of higher mathematics is based on prime numbers. But there's also cryptography, in which prime numbers have a critical importance because really large numbers possess a particularly valuable characteristic. There's no quick, easy way to tell if they're prime numbers or composite numbers, he says.

The difficulty of discerning between huge prime numbers and huge composite numbers makes it possible for a cryptographer to come up with huge composite numbers that are factors of two really big prime numbers, composed of hundreds of digits.

"Imagine that the lock on your door is a 400-digit number," Zegarelli says. "The key is one of the 200-digit numbers that was used to create that 400-digit number. If I've got one of those factors in my pocket, I've got the key to the house. But if you don't have those factors, it's pretty darn tough to get in."

That's why mathematicians have continued to labor to come up with increasingly bigger prime numbers, in an ongoing project called the Great Internet Mersenne Prime Search. In 2018, that project led to the discovery of a prime number that consisted of 23,249,425 digits, enough to fill 9,000 book pages. It took 14 years of computations to come up with that gigantic prime number.

You can imagine how impressed Euclid might have been by that.

This article was updated in conjunction with AI technology, then fact-checked and edited by a HowStuffWorks editor.