show-that-the-set-of-prime-numbers-is-infinite-

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

s h o w t h a t t h e s e t o f p r i m e n u m b e r s

i s i n f i n i t e

Answered by MJS_new last updated on 05/Dec/20

suppose the number of primes is finite and is n

let N = 1 + \underset{j = 1}{\prod^{n}} p_{n}

\Rightarrow \forall n : p_{n} ∤ N \Rightarrow \exists p_{n + 1} : p_{n + 1} ∣ N

\Rightarrow the number of primes is infinite

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