# Thread: Our Only Even Prime Number

1. ## Our Only Even Prime Number

We know that 2 is our only even prime number. My question is WHY?

2. ## Re: Our Only Even Prime Number

Hellio, nycmath!

We know that 2 is our only even prime number.
My question is WHY?

If you know the definition of a Prime Number,
. . the reason should be obvious.

Consider the other even numbers: .$\displaystyle 4,\,6,\,8,\,10,\,\hdots$

All of them are composite numbers: .$\displaystyle (2\!\cdot\!2),\,(2\!\cdot\!3),\,(2\!\cdot\!4),\,(2 \!\cdot\!5),\, \hdots$

3. ## Re: Our Only Even Prime Number

Originally Posted by nycmath
We know that 2 is our only even prime number. My question is WHY?
If you want to think like a mathematician, then this an impossible question. You CAN'T know something unless you know WHY.

4. ## Re: Our Only Even Prime Number

Originally Posted by nycmath
We know that 2 is our only even prime number. My question is WHY?
A prime number is a natural number that only has 1 and itself as factors.

The only factors of 2 are 1 and 2, thus 2 is prime.

Every other even number as a factor of 2 (this is the definition of being even), thus is not prime.

So 2 is the only even prime.

5. ## Re: Our Only Even Prime Number

Just to extend the thinkung a bit - just as 2 is the only multiple of 2 that is a prime number, so also is 3 the only multiple of 3 that is prime, 5 is the only multiple of 5 that is prime, 7 is the only multiple of 7 that is prime, etc.

6. ## Re: Our Only Even Prime Number

Originally Posted by Prove It
A prime number is a natural number that only has 1 and itself as a factor.
With that definition, isn't 1 a prime number?

7. ## Re: Our Only Even Prime Number

Originally Posted by Plato
With that definition, isn't 1 a prime number?
A prime must have exactly two positive divisors - 1 and the number itself. The number 1 has only one divisor, and hence is not priime.

8. ## Re: Our Only Even Prime Number

Originally Posted by ebaines
A prime must have exactly two positive divisors - 1 and the number itself. The number 1 has only one divisor, and hence is not priime.
Well yes, you do quote the correct definition. That was not at all my point, go back and carefully read my post including the quote.
I quoted the post that said "An positive integer is prime if it has only one and itself as factors". That definition makes one a prime. Now it is common to use that definition; I once heard Keith Devlin use it in a talk.

9. ## Re: Our Only Even Prime Number

Very interesting replies. Thank you everyone.