Why $n=2$ should be a prime number however it is even integer and is not similar with other primes?

Question

It is well known that there are infinity many primes, the integer $n=1 $ was a prime number and had been deleted from the list of primes because $1$ has one divisor which is $1$ it self, I would like to ask the similar question which states:

Question:

Why the integer $n=2$ is considerable to be a prime number however is even integer and is not similar to others primes which are odd?

Note: I would like to know a rigourous proof show that because the definition of prime number is not really enough to me with $n=2$ is different from other primes.

Thank you for any help

Answer 1

For a number to be "even" just means that it is divisible by $2$. In much the same way, I could declare that a number is "threeven" if it is divisble by $3$. Then $3$ is the only threeven prime. Do you object to $3$ being prime? Evenness is only special in the fact that it happens to be an important property of numbers so often.

Answer 2

Primes have the special property that, supposing $p$ is prime, and $p$ divides a positive integer $n$, then no matter how we choose to factorise $n=ab$ into positive integers $a$ and $b$, then $p$ must also divide at least one of $a$ or $b$. This holds equally well for the number $2$ as it does for any other odd prime.

Example: $p=3$, and $n=24$ say. We can write $24=1\cdot 24, 2\cdot 12, 3\cdot 8, 4\cdot 6$, and then reversing the order. You can see the claimed property is true.

Example: consider what happens with $4$, and a number it divides like $12$, then we can write $12=1\cdot 12, 2\cdot 6, 3\cdot 4$ and the reverse order, but we immediately see the problem $12=2\cdot 6$, so four cannot be prime, which we knew, but just as an example that $4$ doesn't have the property.

$1$ is also special in that it also divides both factors in the factorisation, which in generalisations is called a 'unit' and so should be excluded from the definition of a prime.