I was teaching my class Euclid's theorem on why there are infinite number of primes. Aside from the idea of proof by contradiction, I wanted to give some more motivation as to why knowing this fact is important.
One idea is that prime numbers are the basis of modern cryptography, and a computer finds it difficult to factor a number composed of two large primes as opposed one with more than two (how much easier actually seems like an interesting problem!). The implication would be that if there are only a finite number of primes in the world, say $N$, then there are $N^2$ numbers that can be used for RSA. Even if N is very large, one could compose a running database of these factorizations and search through them to factor future messages. Therefore there would only be $N^2$ possible internet transactions, which would be a disaster for much of public infrastructure like banking and eCommerce.
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