TITLE: Is there a high-concept explanation for why characteristic 2 is special?
QUESTION [74 upvotes]: The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for $2$. So in these examples characteristic $2$ is a messy special case.
On the other hand, certain types of combinatorial questions can be reduced to linear algebra over $\mathbb{F}_2,$ and this relationship doesn't seem to generalize to other finite fields. So in this example characteristic $2$ is a nice special case.
Is anything deep going on here? (I have a vague idea here about additive inverses and Fourier analysis over $\mathbb{Z}/2\mathbb{Z}$, but I'll wait to see what other people say.)

REPLY [11 votes]: Here is my computational reason (instead of a high-concept explanation) why the primes 2 and 3 are special (hypotheses of many theorems on algebraic groups, linear or projective exclude both these primes). 
The numbers 2 and 3  got into the Primes Club by 'dubious' means!
Given $p$, to certify it as a prime a number, we  need to check no number $d $ with $1 < d \leq [ \sqrt p ]$ divides it. For 2 and 3 this condition is vacuously true as there are no integers in that interval, wheres 5 onwards they really needed to pass the test!