A long long time ago, we found natural numbers can be useful in our life, and some of them seem special. These numbers' multiplication can be a unique identifier for a certain number. We called them: prime. A prime has two factors: 111 and itself(notice that we limit the scope in N\NN). A question comes: Does there have infinite primes?

Proof

Suppose that p1<p2<...<pnp_1 < p_2 < ... < p_np1​<p2​<...<pn​ are all the primes. Let P=p1×p2×...×pn+1P = p_1 \times p_2 \times ... \times p_n + 1P=p1​×p2​×...×pn​+1 and let ppp be a prime dividing PPP; then ppp can't be any of p1,p2,...,pnp_1, p_2, ..., p_np1​,p2​,...,pn​, otherwise ppp would divide the difference P−p1×p2×...×pn=1P - p_1 \times p_2 \times ... \times p_n = 1P−p1​×p2​×...×pn​=1, which is impossible. Therefore, this prime p is not in p1,p2,...,pnp_1, p_2, ..., p_np1​,p2​,...,pn​, and p1,p2,...,pnp_1, p_2, ..., p_np1​,p2​,...,pn​ would not be all the primes.

Conclusion

For a problem related to infinite, this proof is crazy simple delightful. The proof by Euclid at 300 BC, an amazing beautiful proof, but this version definitely not what ancient Greeks would write out since they view numbers in a more concrete way! XD! Here is an English version of Euclid origin words. I hope you enjoy the proof and have a nice day!