[Turkmath:1594] havayı biraz değiştirelim... yet another proof:

[yilmaz akyildiz]

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that there are infinitely many primes
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Posted: 30 Oct 2016 09:41 PM PDT

Sam Northshield [1] came up with the following clever proof that there are
infinitely many primes.

Suppose there are only finitely many primes and let *P* be their product.
Then

[image: 0 < \prod_p \sin\left( \frac{\pi}{p} \right) = \prod_p
\sin\left(\frac{\pi(1+2P)}{p} \right) = 0]

The original publication gives the calculation above with no explanation.
Here’s a little commentary to explain the calculation.

Since prime numbers are greater than 1, sin(π/*p*) is positive for every
prime. And a finite product of positive terms is positive. (An infinite
product of positive terms could converge to zero.)

Since *p* is a factor of *P*, the arguments of sine in the second product
differ from those in the first product by an integer multiple of 2π, so the
corresponding terms in the two products are the same.

There must be some *p* that divides 1 + 2*P*, and that value of *p*
contributes the sine of an integer multiple of π to the product, i.e. a
zero. Since one of the terms in the product is zero, the product is zero.
And since zero is not greater than zero, we have a contradiction.
​ ​

​
(john D. Cook)​

* * *

[1] A One-Line Proof of the Infinitude of Primes, *The American Mathematical
Monthly*, Vol. 122, No. 5 (May 2015), p. 466
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