Why is two the only even number that is prime?

The other prime numbers are all odd numbers such as $5, 11, 127,$ and $37$. So, why is $2$ the only prime even number there is? Maybe it’s because it only has 1 and itself that way, even though it’s even? Check it out on this excellent math page one-fourth from the bottom.

Just scroll down to right there to see. It shows which numbers to 14 are prime and composite.

Well, I hope I’m on the right track using all of you here with me any time! I hear such excitement out there about to type your answers in for me!

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A (positive) even number is some number $n$ such that $n = 2 \cdot k$ for some (positive) integer $k$. A prime number has only itself and $1$ as (positive) divisors.

What happens if $n \not = 2$ in our definition of even numbers?

Why is two the only even $($binary$)$ number that is prime?

For the same reason that three is the only ternary number that is prime. Which is the same reason for which five is the only quinary number that is prime. Etc.

Pick a prime $p$. Call a number $n$ $p$-divisible if $p\mid n$. Then $p$ is the only $p$-divisible prime, trivially. In particular, $2$ is the only $2$-divisible, or even, prime.

In the integers, $-2$ is another even prime.

For variety, in the Gaussian integers, $2$ is not prime: e.g. factors as $(1+i)(1-i)$. The even primes of the Gaussian integers are $\pm 1 \pm i$, although these are all the “same” prime in the same sense that in the integers, $2$ and $-2$ are the “same” prime.

(I define “even” in a number field to be equivalent to its norm being even)

In the ring of all rational numbers with odd denominator, $2/7$ is an even prime. In fact, $2/n$ is prime for every odd integer $n$. (but again, these are all the “same” prime)

There are also number rings that have distinct even primes that are not the “same” in the sense implied above.

A prime number is such that it is divisible by only itself and one. Including 1 as a prime number would violate the fundamental theory of arithmetic, so in modern mathematics it is excluded. Two is a prime because it is divisible by only two and one. All the other even numbers are not prime because they are all divisible by two. That leaves only the odd numbers. Of course, not all odd numbers are prime (e.g. nine is divisible by three).

Because every even number other than 2 is obviously divisible by 2 and so by definition cannot be prime.

The number $2$ has only two whole number factors, $1$ and itself. That’s pretty much it after this: The other numbers that are even up from two are all divisible by that number in some way. This is also known as the “oddest prime” because it’s the only prime number that’s even, so it’s also known as the odd one out. I guess now that that’s pretty much it going to the question about why this can happen.

The word prime comes from the Latin word primus which means “first.” Two (2) is the first even number. In other words, it starts all the even numbers. There is more than one odd prime number because odd numbers are never divisible by 2.