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LostInParadise's avatar

Were you taught Euclid's proof that there are infinitely many primes?

Asked by LostInParadise (32162points) August 20th, 2023

It is such a pretty proof that it should be taught on aesthetic grounds alone. Here it is in all its glory.

We are going to prove that given any finite list of primes, we can always find an additional prime not in the list. Given a list of primes p1, p2,...,pn, Let P = (p1*p2*...*pn)+1. P may be a prime, in which case we are done, since P is greater than all the p values, and so can’t be equal to any of them. If P is not a prime then it must be divisible by primes. P is not divisible by any of p1, p2,... ,pn since P divided by any of them leaves a remainder of 1. Therefore all the primes that divide P must be different from p1,p2,..., pn. QED

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9 Answers

elbanditoroso's avatar

I remember learning this – probably some college class – definitely not high school. Honestly, when I learned this it seemed obvious, and I was surprised that the prof made such a big thing of it.

LostInParadise's avatar

Do you think you could have come up with the proof on your own?

elbanditoroso's avatar

No way. But the concept, yes.

Zaku's avatar

Probably, but I don’t specifically remember. I’m having some unpleasant flashbacks to university honors calculus, which was endless proofs.

elbanditoroso's avatar

@LostInParadise what’s the difference between a theorem and a proof?

LostInParadise's avatar

Theorem is a statement of a mathematical fact that is considered to be useful. For any theorem there are one or more proofs of why it is true.

Strauss's avatar

I think I remember that my high school calculus teacher demonstrated this, informally. The concept seemed obvious, but the proof is absolutely beautiful! And…no, I dont think I could have come up with it on my own.

Forever_Free's avatar

Yes I was taught this. I also worked with a brilliant financial application called Euclid in the EU marketplace.
I also have gone through all 13 books of Euclid’s Elements. There is a fascinating book done by Oliver Byrne

LostInParadise's avatar

That book seems interesting, but according to the Amazon reviews labels for “f” and “s” are sometimes interchanged.

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