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

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

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{they}\:\mathrm{are}\:\mathrm{infinitely}\:\mathrm{many} \\ $$$$\mathrm{primes} \\ $$

Answered by MJS last updated on 03/Feb/20

if there are n primes, p_1 , p_2 , ... p_n ; n∈N let q=1+Π_(j=1) ^n p_j p_j ∤q; 1≤j≤n ⇒ ⇒ q is prime ∨ p∣q with p≠p_j ; 1≤j≤n ⇒ there are at least n+1 primes now let q=1+Π_(j=1) ^(n+1) p_j ...

$$\mathrm{if}\:\mathrm{there}\:\mathrm{are}\:{n}\:\mathrm{primes},\:{p}_{\mathrm{1}} ,\:{p}_{\mathrm{2}} ,\:…\:{p}_{{n}} ;\:{n}\in\mathbb{N} \\ $$$$\mathrm{let}\:{q}=\mathrm{1}+\underset{{j}=\mathrm{1}} {\overset{{n}} {\prod}}{p}_{{j}} \\ $$$${p}_{{j}} \nmid{q};\:\mathrm{1}\leqslant{j}\leqslant{n}\:\Rightarrow \\ $$$$\Rightarrow\:{q}\:\mathrm{is}\:\mathrm{prime}\:\vee\:{p}\mid{q}\:\mathrm{with}\:{p}\neq{p}_{{j}} ;\:\mathrm{1}\leqslant{j}\leqslant{n} \\ $$$$\Rightarrow\:\mathrm{there}\:\mathrm{are}\:\mathrm{at}\:\mathrm{least}\:{n}+\mathrm{1}\:\mathrm{primes} \\ $$$$\mathrm{now}\:\mathrm{let}\:{q}=\mathrm{1}+\underset{{j}=\mathrm{1}} {\overset{{n}+\mathrm{1}} {\prod}}{p}_{{j}} \:… \\ $$

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

$$\mathrm{thanks}\:\mathrm{sir},\mathrm{very}\:\mathrm{clear} \\ $$

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?

$$\mathrm{you}'\mathrm{re}\:\mathrm{welcome} \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{the}\:\mathrm{classical},\:\mathrm{the}\:\mathrm{famous}\:\mathrm{proof},\:\mathrm{I}\:\mathrm{forgot} \\ $$$$\mathrm{who}\:\mathrm{found}\:\mathrm{it},\:\mathrm{maybe}\:\mathrm{Gauss}? \\ $$

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