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Question Number 3275 by Filup last updated on 09/Dec/15
Prove to me that 2 is the only even prime number
$$\mathrm{Prove}\:\mathrm{to}\:\mathrm{me}\:\mathrm{that}\:\mathrm{2}\:\mathrm{is}\:\mathrm{the}\:{only}\:\mathrm{even} \\ $$$$\mathrm{prime}\:\mathrm{number} \\ $$
Commented by Filup last updated on 09/Dec/15
For prime P∈P if P=(x/y) ∴y≠2 ∴x∉E ∵All evens are divisible by two What other methods of proof are there?
$$\mathrm{For}\:\mathrm{prime}\:{P}\in\mathbb{P} \\ $$$$\mathrm{if}\:\:{P}=\frac{{x}}{{y}} \\ $$$$\therefore{y}\neq\mathrm{2} \\ $$$$\therefore{x}\notin\mathbb{E} \\ $$$$\because{All}\:\mathrm{evens}\:\mathrm{are}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{two} \\ $$$$ \\ $$$$\mathrm{What}\:\mathrm{other}\:\mathrm{methods}\:\mathrm{of}\:\mathrm{proof}\:\mathrm{are}\:\mathrm{there}? \\ $$
Answered by 123456 last updated on 09/Dec/15
definition: prime number have only two divisors theorem: all positive even number ≠2 is not prime proof: suppose by absurd that there a even number x≠2 that are prime, we have 1∣x and x∣x, however since x is even, there k∈N x=2k so 2∣x and k∣x so its have the divisors {1,2,k,x} since it has two divisor, so we have x=1∨x=2∨x=k since x≠2, x=1∨x=k if x=k⇒k=2k⇒k=0, but ∀r∈N/{0},r∣x=0, so its have more than 2 divisorsu if x=1⇒x is not even in any case we have a absurd so its not prime □ ps: i dont sure if above proof is correct but note that x>2, x is even x=2k (k>1) so 2∣x,k∣x so its cannot be prime
$$\mathrm{definition}:\:\mathrm{prime}\:\mathrm{number}\:\mathrm{have}\:\mathrm{only} \\ $$$$\mathrm{two}\:\mathrm{divisors} \\ $$$$\mathrm{theorem}:\:\mathrm{all}\:\mathrm{positive}\:\mathrm{even}\:\mathrm{number}\:\neq\mathrm{2}\:\mathrm{is}\:\mathrm{not}\:\mathrm{prime} \\ $$$$\mathrm{proof}: \\ $$$$\mathrm{suppose}\:\mathrm{by}\:\mathrm{absurd}\:\mathrm{that}\:\mathrm{there}\:\mathrm{a}\:\mathrm{even} \\ $$$$\mathrm{number}\:{x}\neq\mathrm{2}\:\mathrm{that}\:\mathrm{are}\:\mathrm{prime},\:\mathrm{we}\:\mathrm{have} \\ $$$$\mathrm{1}\mid{x}\:\mathrm{and}\:{x}\mid{x},\:\mathrm{however} \\ $$$$\mathrm{since}\:{x}\:\mathrm{is}\:\mathrm{even},\:\mathrm{there}\:{k}\in\mathbb{N} \\ $$$${x}=\mathrm{2}{k} \\ $$$$\mathrm{so}\:\mathrm{2}\mid{x}\:\mathrm{and}\:{k}\mid{x} \\ $$$$\mathrm{so}\:\mathrm{its}\:\mathrm{have}\:\mathrm{the}\:\mathrm{divisors} \\ $$$$\left\{\mathrm{1},\mathrm{2},{k},{x}\right\} \\ $$$$\mathrm{since}\:\mathrm{it}\:\mathrm{has}\:\mathrm{two}\:\mathrm{divisor},\:\mathrm{so}\:\mathrm{we}\:\mathrm{have} \\ $$$${x}=\mathrm{1}\vee{x}=\mathrm{2}\vee{x}={k} \\ $$$$\mathrm{since}\:{x}\neq\mathrm{2},\:{x}=\mathrm{1}\vee{x}={k} \\ $$$$\mathrm{if}\:{x}={k}\Rightarrow{k}=\mathrm{2}{k}\Rightarrow{k}=\mathrm{0},\:\mathrm{but}\:\forall{r}\in\mathbb{N}/\left\{\mathrm{0}\right\},{r}\mid{x}=\mathrm{0},\:\mathrm{so}\:\mathrm{its}\:\mathrm{have}\:\mathrm{more}\:\mathrm{than}\:\mathrm{2}\:\mathrm{divisorsu} \\ $$$$\mathrm{if}\:{x}=\mathrm{1}\Rightarrow{x}\:\mathrm{is}\:\mathrm{not}\:\mathrm{even} \\ $$$$\mathrm{in}\:\mathrm{any}\:\mathrm{case}\:\mathrm{we}\:\mathrm{have}\:\mathrm{a}\:\mathrm{absurd} \\ $$$$\mathrm{so}\:\mathrm{its}\:\mathrm{not}\:\mathrm{prime}\:\Box \\ $$$$\mathrm{ps}:\:\mathrm{i}\:\mathrm{dont}\:\mathrm{sure}\:\mathrm{if}\:\mathrm{above}\:\mathrm{proof}\:\mathrm{is}\:\mathrm{correct} \\ $$$$\mathrm{but}\:\mathrm{note}\:\mathrm{that} \\ $$$${x}>\mathrm{2},\:{x}\:\mathrm{is}\:\mathrm{even} \\ $$$${x}=\mathrm{2}{k}\:\:\left({k}>\mathrm{1}\right) \\ $$$$\mathrm{so} \\ $$$$\mathrm{2}\mid{x},{k}\mid{x} \\ $$$$\mathrm{so}\:\mathrm{its}\:\mathrm{cannot}\:\mathrm{be}\:\mathrm{prime} \\ $$

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