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

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 + 2P, 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.

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