Are all prime numbers greater than 2 odd?

Question

Are all prime numbers greater than 2 odd?

I wasn't allowed to assume this was true.

AFSOC that there exists an even prime $n$. Then by definition of even, $n = 2q$, where $q$ is a positive integer. But if $n$ is even, it is divisible by $2$, hence that contradicts $n$'s prime-ness. QED.

What? Is there some wacky prime that can't be assumed odd?

Answer 1

Yes, all prime numbers greater than $2$ are odd.

That's because any even number greater than $2$ can be expressed as $2$ times a number greater than $1$,

so it is composite (not prime).

(As noted in comments to the question, $2$ itself is prime.)

Answer 2

That is correct, all prime numbers in $\mathbb Z$ which are greater than 2 are odd. Likewise all prime numbers less than $-2$ are also odd.

A prime number is a non-unit number (that is, not $-1$ or 1) which is divisible only by itself, the units and the associates of itself (the number times a unit). So, for example, $-7$ is prime, 14 is not.

And no, there is no wacky prime that can't be assumed odd. Except maybe $-2$ and 2. Those two are kind of wacky, in my opinion.

The thing is that in math (or "maths," if you prefer), if you haven't proven an assertion, that assertion might be false. If you make an assertion without proof, someone might challenge you to prove it.

Fortunately, the assertion that all primes greater than 2 are odd is an easy assertion to prove. So, if anyone challenges you on that, you can just quickly state the proof and move on.

Answer 3

Yes---

For if they they weren't, then they would be even; and hence, divisible by $2$; and hence, not prime.