Here's why we still don't understand prime numbers

Prime numbers are one of the most basic topics of study in the branch of mathematics called number theory.

Primes are numbers that can only be evenly divided by themselves and one. For example, 7 is a prime number, since I’m left with a remainder or a fractional component if I divide seven by anything other than itself or one. Six is not a prime because I can divide 6 by 2 and get 3.

One of the reasons primes are important in number theory is that they are, in a certain sense, the building blocks of the natural numbers. The fundamental theorem of arithmetic (the name of which indicates its basic importance) states that any number can be factored into a unique list of primes. 12 = 2 x 2 x 3, 50 = 5 x 5 x 2, 69 = 3 x 23.

Studying numbers, then, basically amounts to studying the properties of prime numbers. Mathematicians have, over the millennia, figured out quite a bit about the prime numbers. One of Euclid’s most famous proofs shows that there are infinitely many primes.

The basic idea of the proof is that if there were only finitely many primes, and we had a list of all of those prime numbers, we could multiply them all together and add 1, creating a new number that isn’t divisible by any of the prime numbers on our list.

That number would either itself be a prime number not on our list, or would have a prime divisor not on our list. Either way, we contradict the idea that there could be a finite list of primes, and so there have to be infinitely many.

In the nineteenth century, mathematicians proved the Prime Number Theorem. Given some large natural number, the theorem gives a rough estimate for how many numbers smaller than the given number are prime. Primes get rarer among larger numbers according to a particular approximate formula.

Despite all the things we know about prime numbers, there are plenty of deceptively simple conjectures about primes that have not yet been either proven or disproven. Here are some of those conjectures.