The whole sine and product business just encodes and obfuscates the claim that "there exists a prime p, such that (1+2product_of_primes) is divisible by p". And

The whole sine and product business just encodes and obfuscates the claim that "there exists a prime p, such that (1+2product_of_primes) is divisible by p". And the two sides of the equation just show how both the claim and its negation can be derived if one assumes a finite set of primes. It's true since any natural number > 1 (including 1+2*product_of_primes) has a prime divisor, and it's false because for any prime p, 2product_of_primes is divisible by p but 1 is not, so the sum cannot be divisible by p. It's actually the classic proof at its core, just with some obfuscation.