Bernhard Riemann's equation unlocks the secrets of prime numbers

"Prime numbers are the atoms of arithmetic," says Marcus du Sautoy of the University of Oxford in the UK. "They are the most basic and important numbers at the heart of the mathematical world. But amazingly, despite over 2000 years of investigation, we still don't understand them at all."

A prime number is only divisible by itself and by 1. For instance, 3 is a prime number, but 8 is not because it can be divided by 2 and 4.

The mathematician Bernhard Riemann published this equation in 1859. It allows you to calculate the number of primes below a given number. For instance, the Riemann equation reveals that there are 25 primes between 1 and 100.

Today all the codes used on the internet exploit the primes to keep messages secure

Riemann's equation reveals that prime numbers are controlled by something called the zeta function, "which at first sight has nothing to do with prime numbers," says du Sautoy.

"For me the equation captures one of the important traits of good mathematics: it tells a story," he says. "The transformation, from one side of the equation describing the primes, to the other controlled by these zeros, is like the discovery of a secret tunnel connecting two parts of the mathematical world that no one thought had anything to do with each other."

This equation implies that there is some deeper rule governing which numbers are prime. "Mathematicians now spend their time trying to understand patterns that seem to be at the heart of these zeros," says du Sautoy.

Prime numbers are of huge practical importance, because most cryptography relies on them. "Today all the codes used on the internet exploit the primes to keep messages secure," says du Sautoy. "Unlock the secret of the primes and it's possible all these codes too will be unlocked."

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