TITLE: Does “evenness” have a deeper mathematical significance than being divisible by 2?
QUESTION [13 upvotes]: I have read and heard many times that “2 is the only even prime number”.
If I was to say “2 is the only prime number divisible by 2” it would be mere tautology; it follows inevitably from the definition of prime numbers, in the same way that 3 is the only prime number divisible by 3, 5 is the only prime number divisible by 5, etc.
Does being even have some deeper mathematical significance than being divisible by 2, or is “2 is the only even prime number” as tautological as “2 is the only prime number divisible by 2”?

REPLY [4 votes]: Very relevant is the Strong law of small numbers:

There aren't enough small numbers to meet the many demands made of them.

In the case of $2$:

There are two few even primes.

One could even say that $2$ is prime simply 'because' it is two small to have a non-trivial factor, being just adjacent to one. Similarly $3$ is prime simply 'because' it is right after $2$. Furthermore, every prime is the only prime divisible by it, simply by definition of prime. So $2$ is not special; $3$ likewise is the only prime divisible by $3$.
Also, see the excellent big list of Examples of Apparent Patterns that Eventually Fail and also a specialized question about the specialness of characteristic $2$.