prove-that-they-are-infinitely-many-primes-

Question Number 80503 by Rio Michael last updated on 03/Feb/20

prove that they are infinitely many

primes

Answered by MJS last updated on 03/Feb/20

if there are n primes, p_{1}, p_{2}, \dots p_{n}; n \in N

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

p_{j} ∤ q; 1 ⩽ j ⩽ n \Rightarrow

\Rightarrow q is prime \lor p ∣ q with p \neq p_{j}; 1 ⩽ j ⩽ n

\Rightarrow there are at least n + 1 primes

now let q = 1 + \underset{j = 1}{\prod^{n + 1}} p_{j} \dots

Commented by Rio Michael last updated on 03/Feb/20

thanks sir, very clear

Commented by MJS last updated on 03/Feb/20

{you}^{'} re welcome

this is the classical, the famous proof, I forgot

who found it, maybe Gauss ?

Commented by Rio Michael last updated on 03/Feb/20

Guess so sir

Commented by mr W last updated on 03/Feb/20

E u c l i d d i d i t!

Commented by MJS last updated on 03/Feb/20

thank you

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