Prove-to-me-that-2-is-the-only-even-prime-number-

Question Number 3275 by Filup last updated on 09/Dec/15

Prove to me that 2 is the o n l y even

prime number

Commented by Filup last updated on 09/Dec/15

For prime P \in P

if P = \frac{x}{y}

∴ y \neq 2

∴ x \notin E

∵ A l l evens are divisible by two

What other methods of proof are there ?

Answered by 123456 last updated on 09/Dec/15

definition : prime number have only

two divisors

theorem : all positive even number \neq 2 is not prime

proof :

suppose by absurd that there a even

number x \neq 2 that are prime, we have

1 ∣ x and x ∣ x, however

since x is even, there k \in N

x = 2 k

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 \lor x = 2 \lor x = k

since x \neq 2, x = 1 \lor x = k

if x = k \Rightarrow k = 2 k \Rightarrow k = 0, but \forall r \in N / {0}, r ∣ x = 0, so its have more than 2 divisorsu

if x = 1 \Rightarrow 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 = 2 k (k > 1)

so

2 ∣ x, k ∣ x

so its cannot be prime

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