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    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, \displaystyle 0<\prod_{p \;\text{prime}}\sin \left(\frac{\pi}{p}\right) =\prod_{p\; \text{prime}}\sin\left(\frac{\pi  }{p}\left(1+2\prod_{q\; \text{prime}} q\right)\right)=0.
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    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:

    \displaystyle \prod_{p\text{ prime}} \dfrac{1}{1-\frac{1}{p}} = \sum_{n>1} \frac{1}{n} = \infty. So the product must have a non-finite number of terms.

    Again, there's a fair bit that really needs better explanation/justification though...
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    (Original post by DFranklin)
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    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 p and claiming irrationality of  \zeta(3).
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    (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:

    \displaystyle \prod_{p\text{ prime}} \dfrac{1}{1-\frac{1}{p}} = \sum_{n>1} \frac{1}{n} = \infty. 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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