Euclid's Greatest Hit: Book IX, Proposition 20

Around 300 BC, a man named Euclid sat down in Alexandria and wrote a textbook. It became the second most printed book in history (after a certain religious bestseller) and stayed in active classroom use for over two thousand years. The book was called the Elements, and buried in its ninth volume is a quiet little proposition that changed mathematics forever.

The Setup

Book IX, Proposition 20 of Euclid’s Elements states, in modern language:

There are more primes than any given finite collection of primes.

Or, as we’d say today: the primes are infinite.

This was not obvious. The ancient Greeks could see that primes get rarer as numbers grow. Between 1 and 100, there are 25 primes. Between 1 and 1,000, only 168. They thin out, becoming harder to find, like wildflowers in an expanding desert. A reasonable person might wonder: do they eventually stop?

Euclid proved they don’t. And he did it in about four sentences.

The Proof (The Real One)

Here’s Euclid’s argument, barely modernized:

Suppose you have some primes — call them p₁, p₂, …, pₙ. (It doesn’t matter how many.)

Multiply them all together and add 1:

N = p₁ × p₂ × … × pₙ + 1

Now look at N. Divide it by any prime on your list. You always get remainder 1 — because N is one more than a multiple of each of them.

So either:

N is itself prime (and it’s not on your list), or
N has a prime factor that’s not on your list (because none of the listed primes divide it evenly)

Either way, your list was incomplete. No matter how many primes you start with, there’s always another one.

That’s it. That’s the proof. No calculus, no algebra, no complex machinery. Just multiplication, addition, and ruthless logic.

A Common Misconception

Many people describe this proof as: “Multiply all known primes and add 1, and the result is a new prime.” That’s not quite right. The number N isn’t always prime — it might be composite. But its prime factors are guaranteed to be new, not already on your list.

For example: 2 × 3 × 5 × 7 × 11 × 13 + 1 = 30,031 = 59 × 509. Not prime! But 59 and 509 are both primes that weren’t in the starting list. Euclid’s argument still works perfectly.

Why It Matters

Proposition 20 is often cited as the first truly great proof in mathematics. It’s beautiful for several reasons:

It’s a proof by contradiction (or more precisely, a proof that any finite list can be extended). The logical structure is airtight.
It uses no advanced tools. A bright twelve-year-old can follow it.
It settles the question permanently. There’s no “probably,” no “under certain conditions.” Primes. Are. Infinite. Done.
It’s constructive (sort of). It doesn’t just say more primes exist — it gives you a recipe for finding a number that reveals a new one.

The Man Himself

We know almost nothing about Euclid as a person. He taught in Alexandria, he wrote the Elements and several other works, and… that’s about it. No birth date, no death date, no dramatic biography. He’s a ghost who left behind thirteen books of geometry and number theory that educated essentially every mathematician for the next two millennia.

There’s an anecdote (possibly apocryphal) that when a student asked Euclid what practical use geometry had, he turned to his servant and said: “Give him a coin, since he must make gain from what he learns.”

Whether or not it happened, it sounds exactly like something a mathematician would say.

Your Infinite Claim

When you buy a prime on A Prime for You, you’re drawing from a well that, thanks to Euclid, we know will never run dry. Your prime is unique, it’s yours, and there are always more where it came from.

Euclid would approve. (Probably. We don’t actually know what he approved of. He’s very mysterious.)