A prime number is a number that is only divisible by one and itself, which is essentially saying that it has no divisor. That takes half of all possible numbers 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. 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, you’ll have no use for this article.
So right out of the gate, we know that prime numbers are important. How could the group that spent hundreds of years obsessing over Fermat’s Last Theorem possibly turn away from a set of numbers that, in a certain sense, contains all other numbers? 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 — 28 is (2 * 14), which is in turn (2 * (2 * 7)). 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). It’s quite another again to find the prime factors of a number fifty digits long. Though the best mathematicians have chewed on the problem for hundreds of years, there just doesn’t seem to be any way to factor large numbers efficiently.
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This little quantum prototype could be the beginning of the end for modern computer encryption.
That fact makes primes vitally important to communications. Most modern computer cryptography works by using the prime factors of large numbers. The large number that was used to encrypt a file can be publicly known and available, because the encryption works so only the prime factors of that large number can be used to decrypt it again. 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.
Primes are of the utmost importance to number theorists because they are the building blocks of whole numbers, and important to the world because their odd mathematical properties make them perfect for our current uses. 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. When researching prime numbers, mathematicians are always being both prosaic and practical.
This story originally appeared on Geek.
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