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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

$$\mathrm{if}\mathrm{there}\mathrm{are}n\mathrm{primes},p_1,p_2,...p_n;n\in \mathbb{N}$$$$\mathrm{let}q=1+\underset{j=1}{\prod ^n}{p}_j$$$$p_j\nmid q;1⩽j⩽n\Rightarrow$$$$\Rightarrow q\mathrm{is}\mathrm{prime}\vee p\mid q\mathrm{with}p\ne p_j;1⩽j⩽n$$$$\Rightarrow \mathrm{there}\mathrm{are}\mathrm{at}\mathrm{least}n+1\mathrm{primes}$$$$\mathrm{now}\mathrm{let}q=1+\underset{j=1}{\prod ^{n+1}}{p}_j...$$

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

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

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

$$\mathrm{Guess}\mathrm{so}\mathrm{sir}$$

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

$$Eucliddidit!$$

Commented by MJS last updated on 03/Feb/20

$$\mathrm{thank}\mathrm{you}$$

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