I didn't even know that this was a misconception in the first place lmao
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This is really interesting, and I actually dug into this a while ago. Technically, the proof is complete after showing none of the original primes can divide the created number, but for human understanding I think it's good to emphasize that the created number can be prime or...
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...if not prime, at least whatever primes divide it cannot be in the original list. Some versions of the proof emphasize that, but some don't, & a reason the proofs are different is that Euclid didn't even really have the language we use for math (so translations take liberties)
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I don't know how many times I've had to repeat this to the gardener.
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But you will get even a bigger prime as a factor, thus a proof that primes are not finite in number.
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Is there any idea of the "rarity" of this occuring? I.e. as you approach infinity, how often would you expect this to be a prime?
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Of course, but a prime that divides that cannot be one of the "first n primes" which means that the list can never contain all of the primes.
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That is not a misconception.
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It is true.
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