What Are Prime Numbers, and Why Are They Important?

If you can only vaguely recall your elementary school math lessons, you might not remember what prime numbers are. Unfortunately, that's a shame, because whether you're securing your emails from hackers or browsing the internet privately on a virtual private network (VPN), you're actually using prime numbers without even realizing it.

Prime numbers play a vital role in RSA encryption, where they are used as keys to decrypt messages buried in digital code. Prime numbers also have numerous other uses in various fields, making it important to grasp their significance. Now, to answer your original question — is 1 a prime number and why are prime numbers important?

What Exactly Is a Prime Number? And How Does 1 Fit In?

So, what exactly are prime numbers? And why have they become so crucial in today's world? According to Wolfram MathWorld, a prime number — or simply a prime — is a positive integer greater than 1 that can only be divided by 1 and itself. In other words, it has exactly two divisors. With this understanding of prime numbers, it becomes clear that the number 1 isn't a prime number.

A useful way to remember this is to recognize that a prime number cannot be divided by any other positive natural number without resulting in either a remainder, a decimal, or a fraction. For example, consider the prime number 13. It only has two divisors: 1 and 13. If you divide 13 by 6, you'll get 2 with a remainder of 1. Dividing any prime number by another natural number always results in a leftover.

Was 1 Ever Considered a Prime Number?

Throughout history, mathematicians have debated what truly qualifies as a prime number. One of the central points of this discussion was whether or not the number 1 should be considered prime. In the 19th century, a debate arose over whether 1 could be classified as a prime number.

At one point, people believed 1 to be a prime number. This belief was based on the idea that a prime number is defined as having exactly two positive divisors: one and the number itself. As a result, 1 seemed to meet the criteria, posing a challenge when trying to categorize it.

However, as mathematics progressed, this viewpoint shifted. To create more consistent and coherent number theories and theorems, mathematicians revisited the definition of prime numbers. This led to a clear distinction between prime and composite numbers.

According to the definition that a prime number must have exactly two distinct positive divisors, 1 did not fit, as it only has one distinct divisor: 1 itself. Consequently, the categorization was updated, and 1 was no longer considered a prime number.

This change ensured that every positive integer greater than 1 would be classified as either prime or composite, providing clarity to mathematical theories and eliminating potential confusion. Although the debate about 1 has largely been settled with the consensus that it is not prime, the historical discussion highlights the evolving nature of mathematical definitions and the ongoing pursuit of precision in the field.

Why Is 2 the Only Even Prime Number?

"The only even prime number is 2," says Debi Mink, a retired associate professor of education at Indiana University Southeast, with a focus on elementary mathematics. "All other prime numbers are odd." This occurs because prime numbers are those with more than two factors. So, let's dive into that concept.

All even numbers are composite, except for 2. This is because 2 has only two factors: 1 and 2 itself, making it a prime number. A prime number is defined by having exactly two distinct factors, which 2 fulfills. Therefore, 2 is a prime number.

Prime numbers include values like 2, 3, 5, 7, 11, 13, and 17, all of which have only two factors: 1 and the number itself. Numbers such as 4, 6, 8, 9, 10, and 12 are not prime since they possess more than two factors.

What distinguishes Prime Numbers from Composite Numbers?

Composite numbers are essentially the opposite of prime numbers. These are numbers that can be divided by other values besides just 1 and the number itself.

Mark Zegarelli, the author of various books on mathematics in the well-known "For Dummies" series, and a teacher of test prep courses, uses an example involving coins to help explain to his students the difference between prime and composite numbers.

"Consider the number 6," says Zegarelli, referring to a composite number. "Imagine having six coins. You could arrange them into a rectangle with two rows of three coins. The same can be done with 8 by arranging four coins into two rows. With 12, you could create multiple rectangles — two rows of six coins or three rows of four coins."

"However, if you take the number 5, no matter how you try, you can't form a rectangle," Zegarelli points out. "The best you can do is line them up in a single row of five coins. You could call 5 a non-rectangular number, but it's simpler to call it a prime number."

There are many other prime numbers — 2, 3, 7, and 11 are just a few, and the list continues indefinitely. The Greek mathematician Euclid, around 300 B.C.E., developed a Proof of the Infinitude of Primes, possibly the first proof that demonstrated the existence of infinitely many prime numbers. In ancient Greece, where the modern concept of infinity wasn't fully grasped, Euclid described the number of primes as "more than any assigned multitude of prime numbers."

Zegarelli suggests another approach to understanding prime and composite numbers by considering them as products of factors. "For example, 2 times 3 equals 6, so 2 and 3 are factors of 6. There are two ways to make six — 1 times 6, and 2 times 3. I refer to these as factor pairs. With composite numbers, there are multiple factor pairs, while with prime numbers, there is only one pair: 1 times the number itself."

Zegarelli argues that proving the infinitude of prime numbers isn't particularly difficult. He explains, 'Imagine if there were a final, largest prime number, which we'll call P. Now, take all the prime numbers up to P and multiply them together. If you then add one to the result, that number must be a prime.'

In contrast, composite numbers are always divisible by smaller prime numbers. 'A composite number might be divisible by other composites as well, but ultimately, it can be broken down into a set of prime numbers.' For example, the number 48 can be factored into two numbers, 6 and 8, but if you break it down further, it becomes 2 times 3 times 2 times 2 times 2.

What is the Sieve of Eratosthenes?

The Sieve of Eratosthenes is an ancient method devised by the Greek mathematician Eratosthenes in the third century B.C.E. It's used to identify both prime and composite numbers from a set of integers.

The Sieve of Eratosthenes operates on the principle that multiples of a prime number cannot be prime themselves. So, while looking for primes, you can eliminate all the multiples of each prime, saving time by preventing unnecessary checks of non-prime numbers.

Prime Numbers Between 1 and 100

Between 1 and 100, there are exactly 25 prime numbers:

Why Prime Numbers Matter

Why have prime numbers captivated mathematicians for millennia? As Zegarelli points out, much of advanced mathematics relies on prime numbers. Additionally, prime numbers play a pivotal role in cryptography, where exceptionally large numbers are valuable because they are hard to categorize as either prime or composite.

The challenge of distinguishing between immense prime numbers and large composite numbers allows cryptographers to generate gigantic composite numbers that are products of two massive prime numbers, each with hundreds of digits.

"Picture your door's lock as a 400-digit number," Zegarelli explains. "The key would be one of the 200-digit numbers used to form that 400-digit lock. If I have one of those numbers, I have the key to your house. But without those numbers, breaking in becomes incredibly difficult."

This is why mathematicians keep striving to uncover ever-larger prime numbers, through a project called the Great Internet Mersenne Prime Search. In 2018, this initiative led to the discovery of a prime number with 23,249,425 digits, enough to fill 9,000 pages of a book. It took 14 years of computations to find this colossal prime number.

One can only imagine the level of astonishment Euclid would have felt upon discovering that.