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Question Number 73536 by Rio Michael last updated on 13/Nov/19
prove that they are infinitely many primes
$${prove}\:{that}\:{they}\:{are}\:{infinitely}\:{many}\:{primes} \\ $$
Answered by MJS last updated on 13/Nov/19
if the number of primes is finite, number  them p_1 , p_2 ,...p_n   now multiply them  q=p_1 p_2 ...p_n   q+1 will not be divisible by any of them ⇒  there′s at least one more prime p_(n+1)   now multiply them all...
$$\mathrm{if}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{primes}\:\mathrm{is}\:\mathrm{finite},\:\mathrm{number} \\ $$$$\mathrm{them}\:{p}_{\mathrm{1}} ,\:{p}_{\mathrm{2}} ,…{p}_{{n}} \\ $$$$\mathrm{now}\:\mathrm{multiply}\:\mathrm{them} \\ $$$${q}={p}_{\mathrm{1}} {p}_{\mathrm{2}} …{p}_{{n}} \\ $$$${q}+\mathrm{1}\:\mathrm{will}\:\mathrm{not}\:\mathrm{be}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{any}\:\mathrm{of}\:\mathrm{them}\:\Rightarrow \\ $$$$\mathrm{there}'\mathrm{s}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{more}\:\mathrm{prime}\:{p}_{{n}+\mathrm{1}} \\ $$$$\mathrm{now}\:\mathrm{multiply}\:\mathrm{them}\:\mathrm{all}… \\ $$
Commented by MJS last updated on 13/Nov/19
example  P={2, 3, 5, 7}  2×3×5×7+1=211 which is prime  2×3×5×7×211+1=44311=73×607  ...
$$\mathrm{example} \\ $$$${P}=\left\{\mathrm{2},\:\mathrm{3},\:\mathrm{5},\:\mathrm{7}\right\} \\ $$$$\mathrm{2}×\mathrm{3}×\mathrm{5}×\mathrm{7}+\mathrm{1}=\mathrm{211}\:\mathrm{which}\:\mathrm{is}\:\mathrm{prime} \\ $$$$\mathrm{2}×\mathrm{3}×\mathrm{5}×\mathrm{7}×\mathrm{211}+\mathrm{1}=\mathrm{44311}=\mathrm{73}×\mathrm{607} \\ $$$$… \\ $$
Commented by Rio Michael last updated on 13/Nov/19
thank you so much sir
$${thank}\:{you}\:{so}\:{much}\:{sir} \\ $$

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