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

Question Number 73536 by Rio Michael last updated on 13/Nov/19

p r o v e t h a t t h e y a r e i n f i n i t e l y m a n y p r i m e s

Answered by MJS last updated on 13/Nov/19

if the number of primes is finite, number

them p_{1}, p_{2}, \dots p_{n}

now multiply them

q = p_{1} p_{2} \dots p_{n}

q + 1 will not be divisible by any of them \Rightarrow

{there}^{'} s at least one more prime p_{n + 1}

now multiply them all \dots

Commented by MJS last updated on 13/Nov/19

example

P = {2, 3, 5, 7}

2 \times 3 \times 5 \times 7 + 1 = 211 which is prime

2 \times 3 \times 5 \times 7 \times 211 + 1 = 44311 = 73 \times 607

\dots

Commented by Rio Michael last updated on 13/Nov/19

t h a n k y o u s o m u c h s i r