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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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5$\begingroup$ 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/… $\endgroup$– Sam NolenCommented May 3, 2013 at 11:44
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$\begingroup$ Actually, because it is so even $\endgroup$– Francesco PolizziCommented May 3, 2013 at 12:57
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$\begingroup$ 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. $\endgroup$– Douglas ZareCommented May 3, 2013 at 21:02
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$\begingroup$ 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? $\endgroup$– The AmplitwistCommented May 6, 2023 at 11:58
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4 Answers
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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 3, 2013 at 11:42
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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 $p=2$.
answered May 3, 2013 at 11:53
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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.
answered May 3, 2013 at 12:09