Depending on how much you're willing to assume and what exactly counts as "one line," it could be as simple as:

> One plus the product of all primes is itself prime, and larger than all primes.

This omits a bunch of stuff, but it seems like less than the linked paper.

> One plus the product of all primes is itself prime

One plus the product of the first N primes is not necessarily prime. Consider 2 * 3 * 5 * 7 * 11 * 13 + 1 = 30031 = 59 * 509

How about: one plus the product of a list of primes has a prime factor not in the list.

Yup, that works!

I like that version better anyway. Thanks for helping me improve it!

Yes! Because primes are infinite! The statement assumes limited primes. And, because it falsifies, it proves primes must be unlimited. I find this argument based on prime factors much simpler.

I would have been fine with "would have to be prime"; I like the edit Mike suggested even better.

We know it can be written out as a haiku: https://xkcd.com/622/!

While that's very clever, it doesn't accurately present Euclid's argument because the haiku is

  Top prime's divisors'
  product (plus one)'s factors are...?
  Q.E.D., bitches!

This doesn't include Euclid's argument about multiplying all of the primes, mistakenly referring instead to "top prime's divisors".

The "top prime's divisors' product" would be equal to the top prime itself, so Randall's haiku asks "if there is a largest prime p, what are the divisors of (p+1)?" which doesn't create any contradiction (it could simply be divisible by various smaller primes!).

Maybe we should amend it to

  Take factorial
  of top prime, then add one: what
  are the divisors?

Couldn't you just change "divisors' product" to "factorial's"? I guess that it does some violence to the metre.

You could work with it and get the original last line back:

  Factorial of
  top prime, plus one: factor that!
  Q.E.D., bitches!

But then you can't exclaim Q.E.D., bitches! and it would be a terrible cartoon. :)

It's restored in a version further down in the thread.

It's not the factorial either, that would include composites in the product too.

Euclid didn't use the factorial in his original proof, but it still produces a logically correct argument and it has fewer syllables. Metri causa. :-)

Using the factorial is better, I think, even if ahistorical; it avoids the slight unpleasantness of having to show that every non-unit integer is divisible by a prime.