Question Number 124691 by mathocean1 last updated on 05/Dec/20
$${show}\:{that}\:{the}\:{set}\:{of}\:{prime}\:{numbers} \\ $$$${is}\:{infinite} \\ $$
Answered by MJS_new last updated on 05/Dec/20
$$\mathrm{suppose}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{primes}\:\mathrm{is}\:\mathrm{finite}\:\mathrm{and}\:\mathrm{is}\:{n} \\ $$$$\mathrm{let}\:{N}=\mathrm{1}+\underset{{j}=\mathrm{1}} {\overset{{n}} {\prod}}{p}_{{n}} \\ $$$$\Rightarrow\:\forall{n}:{p}_{{n}} \nmid{N}\:\Rightarrow\:\exists{p}_{{n}+\mathrm{1}} :{p}_{{n}+\mathrm{1}} \mid{N} \\ $$$$\Rightarrow\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{primes}\:\mathrm{is}\:\mathrm{infinite} \\ $$