The infinite product of 2×2×2×2×2×2×2... = 0

If we set p = 2×2×2×2×2×2×2... we can see that 2×2×2×2×2×2×2... = 2×p

Leading to p = 2×p

Subtract p from both side; 0 = p

Giving 2×2×2×2×2×2×2... = 0

Sadly the Mersenne number 2^∞-1 = -1, and therefore isn't prime.