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Question Number 124691 by mathocean1 last updated on 05/Dec/20

$$showthatthesetofprimenumbers$$$$isinfinite$$

Answered by MJS_new last updated on 05/Dec/20

$$\mathrm{suppose}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{primes}\mathrm{is}\mathrm{finite}\mathrm{and}\mathrm{is}n$$$$\mathrm{let}N=1+\underset{j=1}{\prod ^n}{p}_n$$$$\Rightarrow \mathrm{\forall }n:p_n\nmid N\Rightarrow \mathrm{\exists }{p}_{n+1}:p_{n+1}\mid N$$$$\Rightarrow \mathrm{the}\mathrm{number}\mathrm{of}\mathrm{primes}\mathrm{is}\mathrm{infinite}$$

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