Here’s something you won’t hear everyday: Prime numbers are cool.
I know what you’re thinking. If you’re like most people you probably have a viscerally negative reaction to the very idea of a mathematic concept. But I promise, we’re not gonna get into scientific proofs or advanced number theory.
To get you to come around to liking prime numbers we are going to stick to some very basic facts about them. And the first fact is simply: prime numbers are cool. As Carl Sagan points out so eloquently in the novel Contact, there’s a certain importance to prime numbers’ status as the most fundamental building blocks of all numbers, which are themselves the building blocks of our understanding of the universe.
In that book, aliens choose to send a long string of prime numbers as proof that their message is intelligent and not natural in origin, since primes are one thing that cannot change due to differences of psychology, lifestyle, or evolutionary history. No matter what an advanced alien life-form looks or thinks like, if it understands the world around it, it almost certainly has the concept of a prime.
But to truly understand the importance of prime numbers, we will have to go deeper.
What are prime numbers again?
Most people are probably familiar with at least the basic idea of prime numbers. For those that need a refresher, however, here it is.
Primes are the set of all numbers that can only be equally divided by 1 and themselves, with no other even division possible. For example, numbers like 2, 3, 5, 7, and 11 are all prime numbers. If you looking for primes then, half of all possible numbers can be taken off the table right away (the evens), along with all multiples of three, four, five, and so on.
It might seem that this would leave no numbers after a certain point, but in fact, we know that there are an infinite number of primes — though they do become less frequent as we go on.
In fact, that’s part of what makes primes so interesting: not only is the number line studded with primes all the way up to infinity, but that whole number line can be produced using nothing but primes. For instance, 12 can be rewritten as (2 * 2 * 3), and both 2 and 3 are primes. Similarly, 155 can be written as (5 * 31).
An extremely complex mathematical proof can assure you that combinations of prime numbers can be multiplied to produce any number at all — though if you can understand that proof, this article, frankly, is not for you.
Why do people care about primes?
In a sense, we can define primes according to this status as a basic-level number: primes are the total set of numbers which are left over when we rewrite all numbers as their lowest possible combination of integers. When no further factoring can be done, all numbers left over are primes.
This is why primes are so relevant in certain fields — primes have very special properties for factorization. One of those properties is that while it is relatively easy to find larger prime numbers, it’s unavoidably hard to factor large numbers back into primes.
It’s one thing to figure out that 20 is (2 * 2 * 5), and quite another to figure out that 2,244,354 is (2 * 3 * 7 * 53,437). You can imagine then how unfathomably hard it might be then to factor a number 50 or even 100 digits long. It’s so hard in fact that even though the best mathematicians have been working at the problem for hundreds of years, there is still no way to efficiently factor large numbers.
While that may sound like a problem, for the uses of prime numbers it is actually an opportunity. Modern encryption algorithms exploit the fact that we can easily take two large primes and multiply them together to get a new, super-large number, but that no computer yet created can take that super-large number and quickly figure out which two primes went into making it.
Though finding those factors is technically only a matter of time, it’s a matter of so much time that we say it cannot be done. A modern super-computer could chew on a 256-bit factorization problem for longer than the current age of the universe, and still not get the answer.
Whether it’s communicating your credit card information to Amazon, logging into your bank, or sending a manually encrypted email to a colleague, we are constantly using computer encryption.
And that means we are constantly using prime numbers, and relying on their odd numerical properties for protection of the cyber-age way of life. It’s no meaningless academic quest, the effort to better understand prime numbers, since virtually all of modern security relies upon the current limitations of that understanding.
It’s possible that new mathematical strategies or new hardware like quantum computers could lead to quicker prime factorization of large numbers, which would effectively break modern encryption. But even once that happens, pretty much anything that computers can easily do without being able to easily undo will be of interest to computer security.
What does it all mean?
There are dozens of important uses for prime numbers. Cicadas time their life cycles by them, modern screens use them to define color intensities of pixels, and manufacturers use them to get rid of harmonics in their products. However, these uses pale in comparison to the fact that they make up the very basis of modern computational security.
Whatever your thoughts are on prime numbers, you use them every single day and they make up an absolutely vital part of our society. All this because they are an irreducible part of the very fabric of the universe.
And that makes prime numbers pretty cool.