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Then, you would need 10^18 bytes, or a million terabytes of storage to store those primes. That isn't practical, so let's scale back to primes around 10^20 (around 10^18 different primes)

Actually, a million terabytes (an exabyte) is really quite feasible. Not on an individual scale, but I would be very surprised if governments (all of them, really) did not have this sort of thing set up.

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