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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 $>2$?

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marked as duplicate by Felipe Voloch, Dan Petersen, Steven Landsburg, Douglas Zare, Mark Grant May 3 '13 at 12:25

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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You might want to consider expanding your question and discussing some specific results that you have in mind. Also, the following is related: mathoverflow.net/questions/915/… –  Sam Nolen May 3 '13 at 11:44
    
Actually, because it is so even –  Francesco Polizzi May 3 '13 at 12:57
    
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 3 '13 at 21:02
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4 Answers 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 $p$.

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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.

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

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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.

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