Refresh and Motivation

Let‘s recap: primes are whole numbers that can only be divided by 1 or the number itself without remainder. For example, 5 is prime (divisors 1 and 5) but 6 is not (divisors 1, 2, 3 and 6).

There are infinite prime numbers but so far there are no efficient algorithms to determine them. Especially, there is no formula to calculate the n-th prime number, neither recursively, i.e. we can calculate a prime if we know the preceding (smaller) primes, nor explicitly, i.e. we can calculate a prime directly without knowing preceding primes.

For example, this makes the famous RSA cryptosystem so secure. Its public key needed for encryption is based on a product of two (very large) prime numbers. If you want to derive the private key needed for decryption you “just” need to determine the prime factors of this product. However, this takes currently so much computing time that RSA is not unlockable in practice.

But what would happen if we would discover a formula that calculates primes immediately? This could also generate very fast approaches for prime factorization which would mean a death sentence for most cryptosystems today. But is it even possible to find a formula for primes?

The Amazing Euler Product

Leonhard Euler is one of the most brilliant mathematicians the world has ever seen. In the 18th century he derived a formula that is called Euler product today. Here we focus on a special case of his trailblazing discovery. Please don’t stop reading even if the next line looks like hieroglyphic at first sight.

Euler product

We translate: the symbol on the left hand side of the equation represents a product. Furthermore, it is an infinite product over all primes, i.e. we need to replace the variable p by all prime numbers and multiply the terms. Let’s write it down to make it clear.

First factors of the Euler product

This means the following: if we calculate the above product inserting ALL prime numbers we obtain the well-defined result pi²/6. That’s awesome and feels like a mystery. Please let me tell you why.

Disruptive Consequences

We know that there are infinite prime numbers but we haven’t any closed and efficient representation (“formula”) for primes. With computing power, we can just determine the largest known prime number. In spite of this, Euler has proven that we obtain the value pi²/6 if we multiply all primes according to the Euler product — although we don’t know all primes!

IMHO, this shows that there is tremendous knowledge about primes that we haven’t discovered so far. If we can calculate the Euler product over the infinite set of primes we should also be able to derive a formula for primes. For example, for special primes closed representations are already known.

This indicates that we must increase efforts in number theoretical research to discover the true nature of primes. And the person who may unriddle this quest will either be celebrated or persecuted.

Outro

I ask myself if such a nerdy topic will attract readers. I am a fan of number theory, however, it’s not my daily business so I do appreciate your comments. Please let me know if you like to learn more about this stuff or mathematics beyond, maybe I am going to write a follow-up.