"Perfect proofs don't exi-"

sure, but where does the last line come from? You need to explain why p divides 1+ 2 prod p'. To do that, you're back to Euclid's argument anyway.

>if the set of primes is finite

laem and ghey

>non-constructive proof
Show me all the primes brainlet.

>all that fancy notation i can't understand

sigh. making me feel all brainlet

The big pi is the same thing as summation notation, just using multiplication instead of addition.

>The original publication gives the calculation above with no explanation.
fucking mathfags. fuck all of you. most of your papers are nigh unreadable, taking hours to digest, only for the reader to realize that it was a waste of time.

If you're looking for a one line proof of the infinitude of primes, you're obviously ready to waste hours. Further, the logic behind the line isn't advanced. I get that it's not obvious to most, but it requires no advanced knowledge.

yeah but i'm talking about math papers in general like if you a have a real need/use for more advanced math knowledge for a computer program or whatever. in programming, a program is considered well-written if it's easy to read and understand, not if you put everything in one line and hide things behind crazy symbols instead of using proper variable names etc

I can definitely see that. However, these papers are not written for the layman to pick them up and understand them. The papers are written so that people within the same field will understand. That being said, sometimes the logic behind some lines are "difficult" even to others in the field. There is a transition period between graduate and post-PHD papers in the math community wherein mathematicians learn to omit elementary and one step logic statements. I get the desire for brevity, but it detracts from understanding occasionally.

For all the brainlets that can't understand the proof:
Firstly, the first inequality arises since every prime p is larger than 1, so that all the sine arguments are in the first quadrant, and hence positive, and hence the product must be positive. It is not 0 because the zeros of sin occur at integer multiples of pi, so that if [math]\sin(\pi/p)[/math] is 0, then [math]\pi[/math] is not irrational, which is a contradiction.

Secodly, sine is [math]2\pi[/math]-periodic, so that adding any integer multiple of [math]2\pi[/math] gives the same value of sine, and obviously a finite product of primes is an integer, so that adding 2 times that times [math]\pi[/math] is like "adding zero".

Thirdly, since [math]1+2\prod p'[/math] is an integer, it must be divisible by a prime, or be prime itself. Since [math]p[/math] runs over all primes, then it divides one of these. Hence, the argument of the sine is an integer multiple of [math]\pi[/math] and so is 0. Since we have a finite product with one of the arguments being 0, the whole product is 0.

Fourthly, the contradiction that arises is [math]0

>perfect
It's just Euclid's proof restated as a contradiction.
It's not even one line (those are clearly two lines).

>since 1+2 prod p' is an integer, it must be divisible by a prime, or be prime itself.

It's clearly larger than, and not divisible by, any of the p'. You're essentially done. Why do you need the rest of it?

This line of reasoning needs proof (hint: it's incorrect)

it's a clever gimmick, let's leave it at that

>It's clearly larger than, and not divisible by, any of the p'
The statement [math]\forall i \in \mathbb{N} \ \ \forall k

>if the set of primes if finite
stopped reading there

Let P be the product of the first i primes. Then the k-th prime, where k

where is the elegance in a one line proof, when you're adding this extra sin stuff, just euclid's theorem by itself is elegant enough

i agree. fetishizing "one line" proofs is pretty dumb, especially when,as in this case, the proof is so short only because the author neglects to explain the crucial steps. the elegance (or not) of a proof depends on the idea, not the way it's written.

>1+2∏p′ is an integer, it must be divisible by a prime, or be prime itself.
Isn't this an argument FOR an infinite set of primes? If you have a finite set of primes, let's say {2, 3}, then equation 2 will be sin(pi*13/2)*sin(pi*13/3) which will be non-zero, which seems to contradict what the proof is saying.
So how does this tie back to disproving that the set of primes is finite?

en.wikipedia.org/wiki/Proof_by_contradiction

Yes, but you're working under the assumption that you have listed all the primes in the product, so the product must contain a number that divides, say, 13, but that means 13 is a prime, and you didnt list it, so your product doesn't contain all primes.
It's an abstract argument that leads to a contradiction, so it doesn't have to make sense with a trivial example.

The product P of the first k primes is divisible by each of the first k primes. 2*P+1 is not divisible by any of the first k primes. I don't know how to make it any clearer to you.

prove it

If you only had a finite number N of primes, then you'd have their product P. Since P < 2P+1, you would have two options:
>2P+1 is a prime
This is impossible because P is greater than any of the N primes.
>2P+1 is not a prime
Then 2P+1 must be divisible by a positive integer other than itself or 1, so eventually we'd have a prime number dividing 2P+1.

Thank you!

a "good" proof doesn't make use of unnecessary shit.
wtf is trigonometry doing here in this arithmetic elementary result?
the main point is that you build a number with your finite set of primes that is not in the set but would be prime because none of them divides it...