The prime number $2$

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Why is 2 so odd?

I have read few books and articles, almost all of them refer that any prime $p>2$. Just wondering why it has to be $>2$?

You might want to consider expanding your question and discussing some specific results that you have in mind. Also, the following is related: http://mathoverflow.net/questions/915/is-there-a-high-concept-explanation-for-why-characteristic-2-is-special — Sam Nolen, May 03 '13 at 11:44
http://mathoverflow.net/questions/15141/why-is-2-so-odd is very similar, and was closed as an exact duplicate of the question Sam Nolen linked. — Douglas Zare, May 03 '13 at 21:02
Reposting a link mentioned in a previous comment so that it appears in the "Linked" questions list: Is there a high-concept explanation for why characteristic 2 is special? — The Amplitwist, May 06 '23 at 11:58

score 4 · Answer 1 · answered May 03 '13 at 11:42

4

There are many possible answers to this question, and it depends a lot on the context, but certainly one of the main reasons is that you cannot distinguish between $1$ and $-1$ modulo $2$, whereas $1 \not\equiv -1 \pmod{p}$ for any other prime $p$.

answered May 03 '13 at 11:42

Tom De Medts

6,514

score 2 · Answer 2 · answered May 03 '13 at 11:36

2

Do you mean: Why is the prime $2$ "special" (the number theorist's nightmare)? One reason for that is that $2$ is the smallest prime.

answered May 03 '13 at 11:36

score 1 · Answer 3 · answered May 03 '13 at 11:53

1

Because $p-1$ (an expression that appears often dealing with primes) equals 1 iff $p=2$.

answered May 03 '13 at 11:53

Rodrigo A. Pérez

3,071

score 1 · Answer 4 · answered May 03 '13 at 12:09

There is an old saying "All primes are odd, but 2 is the oddest of all!". For example, if only primes $p > 2$ divide the order of some finite group, by the Odd Order Theorem you already know that the group is solvable. If also 2 divides the order, you need more information to draw a conclusion.

The prime number $2$

4 Answers4