The statement that "$2$ is the only even prime number" has always struck me as very peculiar. I do not find this statement mathematically interesting, though I do find the fact that it is presented as something interesting about $2$ or prime numbers to be itself quite interesting.
I find it interesting because this statement has secured its place as a "math-tidbit", if you will, solely because we happen to have a word for being divisible by $2$.
In other words, it is not at all clear to me why the statement "$2$ is the only even prime" is any more interesting or worth saying than the statement "$7$ is the only prime that is divisible by $7$."
Every prime $p$ is the only prime divisible by $p$. This immediately follows from the definition. So my question is, what makes the $2$ case particularly special or interesting, other than the fact that we have a word for divisibility by $2$?
Best Answer
You are right that the quoted statement as such loses its charm upon a moment of consideration. After all, as you say, "$2$ is the only even prime" is not more or less surprising than "$7$ is the only prime divisible by $7$".
One could think of it this way: Why is $2$ the only prime $p$ such that we have a special word for "divisible by $p$"? -- That is a question more about language than about math, but it hints at the fact that the number $2$ holds a more special place in our psychology than all other natural numbers (although some of the other primes, especially $3, 5, 7,$ and $13$, stand out too, in some dominant cultures. $57$, not so much.)
Now, this site is not about psychology, culture, or language. But even mathematicians will sometimes utter a statement similar to the one you quote. What could they mean?
It turns out that in various theories, the prime $p=2$ (and to a lesser but notable extent, $p=3$) behaves differently than all the other primes! See: https://mathoverflow.net/q/160811/27465, and compare https://mathoverflow.net/q/915/27465. Also, the whimsically phrased Why are even primes notable?. A basic example that sets aside $p=2$ (which causes many others) is that there are two rational solutions to $x^2=1$, while for all other $p$, the equation $x^p=1$ has only one real solution. (For more advanced people: The unit group of $\mathbb Z$ is $ \simeq\mathbb Z/2$. Or: The only primitve roots of unity in $\mathbb R$ are $\pm 1$.)
Now the statement "$2$ is the only even prime" might be an exceptionally bad way to express this phenomenon. More fitting might be the old joke, as per Gottfried Helms' comment: "$2$ is the oddest prime."