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Maths in a minute: How many primes?

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Maths in a minute: How many primes?

Euclid

Euclid of Alexandria

Primes are natural numbers that are only divisible by themselves and 1. The first few examples are 2, 3, 5, 7, and 11. And the good news is that they go on forever. No matter how far you move up the number line there'll always be another prime ahead of you. The fact that there are infinitely many primes was already known to the ancient Greeks. Here's a proof, attributed to the most famous Greek mathematician of them all, Euclid (although he didn't use quite the same language and concepts as we do today).

Suppose that there are only finitely many primes. We can label them p1, p2, p3, etc, up to pn, the very last prime. Now define the number P=p1×p2×p3...×pn+1. As an example, if there were only five primes p1=2, p2=3, p3=5, p4=7, and p5=11, then we'd have P=2×3×5×7×11+1=2311. Now, if P is itself a prime number (as it is in our example), then clearly it can't be one of the primes in our list: that's because it's bigger than all of them. If P is not prime, then, like any other natural number, it can be written as a product of primes. Let's pick one of the primes that divides P and denote it by p. Now p cannot be any of the primes p1 up to pn because if it was, then it would divide P(p1×p2×p3...×pn)=1, which is impossible because 1 is not divisible by any natural number. Therefore the collection p1, p2, p3, etc, up to pn cannot contain all of the primes as we had assumed. This contradiction means that there must be infinitely many primes.

You can read more about prime numbers and number theory on Plus.

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