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# One-line proof that there are infinitely many primes watch

1. Just came across this and thought the maths community would appreciate! Due to Sam Northshield, 2015 — unlike the other one-line proofs it uses nothing fancy. In some sense it is just a clever re-phrasing of Euclid but I like it.

Pf. If there are finitely many primes, .
2. Not sure about this one; as you say, it's essentially a re-phrasing of Euclid, but there's so much elided in the 2nd step that you basically have to know the Euclid proof to see it works.

I quite like this one:

. So the product must have a non-finite number of terms.

Again, there's a fair bit that really needs better explanation/justification though...
3. (Original post by DFranklin)
...
Yes I just thought it was quite a neat way of putting it. I like yours too. It can be made ridiculously overkill by cubing the and claiming irrationality of .
4. (Original post by DFranklin)
Not sure about this one; as you say, it's essentially a re-phrasing of Euclid, but there's so much elided in the 2nd step that you basically have to know the Euclid proof to see it works.

I quite like this one:

. So the product must have a non-finite number of terms.

Again, there's a fair bit that really needs better explanation/justification though...
Ahhh me likey this one.

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