Why are even primes notable?

Question

There are much-discussed theorems like Fermat's theorem on sums of two squares which make statements about odd primes only. This makes $2$ seem to be a "special" prime. In their book The book of numbers, Conway and Guy accordingly state that "Two is celebrated as the only even prime, which in some sense makes it the oddest prime of all."

On the other hand, the fact that $2$ is the only even prime is completely trivial, because the term "even" means the same thing as "divisible by $2$" and every prime number has the property that it is the only prime which is divisiable by itself.

So my question is: Is there really something special about even primes and if yes, what is it? Does aesthetics with regard to the theorems we are looking for play a role or is there a mathematical reason? Do we have theorems about primes which aren't divisible by $3, 5, ... $ or are there only results which don't apply to even primes?

Edit: As the user A.G has mentioned in a comment below, in many cases where we have a regular pattern, the fact that $2$ is too small for the pattern to kick in yet seems to be the decisive thing. So in these cases, the notable thing is not that $2$ is the only even prime but that it is the smallest prime.

Answer 1

The quip about "2 being the only even prime" is a bit silly, as you say, since 3 is the only divisible-by-3 prime, etc. For that quip, it's just that parity (odd-or-even) exists in the ambient language.

For $p$ prime, the $p$th roots of unity are in $\mathbb Q$ for $p=1$. Similarly, the $p$th roots of unity lie in all finite fields (of characteristic not $p$...) only for $p=2$.

Quadratic forms and bilinear forms behave differently in characteristic two.

Groups $SL(n,\mathbb F_q)$ do not assume their general pattern yet for small $n$ and $q=2$.

The index of alternating groups in symmetric groups is $2$.

Subgroups of index $2$ are normal.

The canonical anti-involution on a non-commutative ring, that reverse the order of multiplication, is of order $2$.

Answer 2

Parity, either every number is one thing or another, is pretty important.

It's true that $3$ is the only prime divisible by $3$ but some of the primes not divisible by $3$ are $\equiv 1 \pmod 3$ and others are $\equiv -1\pmod 3$ whereas all primes other then $2$ are odd.

If $p<q$ are two different primes then $p+q$ is odd only if $p=2$ but $p+q$ could be any divisibility of $3$. ($3|p+q$ if $p\ne 3$ and $p\equiv -q\pmod 3$. $p+q\equiv 1$ if $p=3$ and $q\equiv 1$ or if $p\equiv q\equiv -1\pmod 3$ and $p+q\equiv -1$ if $p=3$ and $q\equiv 1$ or if $p\equiv q\equiv 1\pmod 3$).

And for $m\le n$ then $p^{m} + p^n = p^m(1+p^n)$ so $p^{m+1}\not \mid p^{m} + p^{n}$ should be a valid result. But... if $p=2$ and $m=n$ then....

Answer 3

Here is a personal view on the 'oddness' of $2$:

Parity is important in a logically dichotomous universe; anything other than nothing in the universe is either $A$ or not-$A$ (for each categorization $A$ of things). As others have pointed out, this linguistic or logical 'ambience' brings $2$ to the forefront of our thinking about many things, even when its properties as a prime are not truly unique.

But $2$ is really unusual among the primes (for me) because it is the only prime (indeed the only positive integer $n>1$) for which $x^n+y^n=z^n$ has integer solutions. I consider this fact to be bafflingly improbable. Why are there solutions (and infinitely many of them) for only one integer exponent, and given that is the case, why is that exponent $2$, rather than another among the infinitude of possible primes?

Answer 4

$2$ indeed is the smallest prime. Another peculiarity is that it supports the "dichotomy" paradigm, that parallels the logical world: true or false, with or without, left or right.

Answer 5

Here's some background to the answer. A group is an order pair of a set and a binary operation ⋅ from that set to itself satisfying the following properties.

• ⋅ is associative
• There is an identity element
• Every element has an inverse with respect to that operation.

One element is said to be an inverse of the other when their product in both orders is the identity element. The order of a group is its cardinality. Also, 2 integers are said to be congruent modulo a certain integer when their difference is a multiple of that integer. For example, 2 is congruent to 6 modulo 4. This is written $2 \equiv 6 \mod 4$. The remainder of 10 on division by 4 is 2 so we can also write $10 \mod 4 = 2$. Some people would write $6 = 10 \mod 4$ to mean $6 \equiv 10 \mod 4$ but this is technically incorrect because it is saying that 6 is equal to $10 \mod 4$. But $10 \mod 4 = 2$ and $6 \neq 2$ so $6 \neq 10 \mod 4$.

2 has the property that not all powers of it have a cyclic multiplication modulo group. No other prime number has that property. Take any positive integer. Now take the powers of one more than it modulo the cube of the starting integer such as powers of 11 modulo 1,000. Taking powers of 11 modulo 1,000, we get 001, 011, 121, 331, 641, 051, 561, 171, 881, 691, 601. In any base x, after the $x^{th}$ power, the first digit is a 1 when the base is odd and is 1 more than half the base when the base is even. This shows that the numbers 1 more than a multiple of x modulo $x^3$ form a cyclic group when ever x mod 4 is 1, 3, or 4 but not when x mod 4 is 2. This is something special about the property of being an even number. Another property such as the property of being a multiple of 3 doesn't have something special about it like that.

The multiplication modulo group of any product of 2 odd primes is never cyclic. That's because you can always find a number that's 1 more than a multiple of one of those primes and 1 less than a multiple of the other one of those primes. It's square must be 1 more than a multiple of the semiprime. For example, $21^2$ is 1 more than a multiple of 55. It turns out that all this comes from the fact that 2 is the only prime number that's so small that any integer that's not a multiple of it is 1 more than a multiple of it, making the multiplication modulo group of any odd prime have order an even number.

A subgroup of a group is a subset that is a group with the operation of the original group. A coset of the group is one where each member can be gotten by right multiplying any member of the coset by a member of the original subgroup. When the product of any member of one coset by any member of another coset always gets you a member of the same coset, the subgroup is said to be a normal subgroup. We can then form the cosets into a group and call it the quotient group. When ever a subgroup has 2 cosets, it is necessarily a normal subgroup. For any other prime number, it is not always the case that a subgroup with that number of cosets is a normal subgroup.