Mathematicians: what's so special about evenness?
From Wikipedia:
In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd. An integer is even if it is divisible by two and odd if it is not even. Fine. But, is there anything more to it? I mean, 2 is the "only even prime number". That's like saying "2 is the only prime number divisible by two". Is that more remarkable than 3 being the only prime number divisible by three? Even integer on number line is always followed by odd integer. But then again, integer divisible by 3 or 7 or n is always followed by nondivisible integer n1 times, so... What I'm clumsily trying to say is: if we ignore cultural and historical references and capability of our brains to handle pattern of smallish numbers and sets, is "being even" meaning anything more than "being divisible by two" from purely mathematical standpoint? 
Sometimes it's relevant that a number is divisible by two. Sometimes it's relevant that a number is divisible by three, or four, or some other number. But the larger the factor, the less common it is for divisibility to be relevant. 2 is the smallest nontrivial factor, so divisibility by 2 is more often relevant than the others.

In terms of just the straightforward even/odd thing it's just a handy way to split the integers into two groups. There's a lot* of ways of doing this, of course. E.g., prime and nonprime.
Superficially even numbers are not more "special" than odd numbers. It turns out that a lot of theorems are going to mention "even" somewhere along the way than they do "odd". It's like how we use the "prime" a lot more than we use "nonprime"/"composite". We're not trying to use one over the other. That's just the way Math rolls. Some properties gets used/referred to more than others. It's an inherent property of these systems. And if you're talking my field, Computer Science, evenness is inherently built into the base of the thing. It all comes down to 2s. * Understatement of the megamillennium. 
I'm sure we can come up with any number of theorems which require one to distinguish between even and odd primes. For example, try investigating the number of distinct solutions to x^{2} = a (mod p), or formulating a quadratic reciprocity theorem.

There are bunches of theorems that start, "Let p be an odd prime, then ..." that fail for two. The standard proof that every number is a sum of four squares (which needs to be proved only for primes) starts out by showing that some multiple kp, with k < p, is a sum of four squares and then takes two different paths depending on whether k is odd or even.

As others have said, sometimes 2 behaves differently from the odd primes and sometimes it doesn't. In my own work, the difference often comes from the fact that, working modulo 2, we always have x = x, which is not true working modulo an odd prime.
Here's a specific case where that makes a difference. I often work with systems that are anticommutative, meaning that there's a multiplication with yx = xy. Looking at the case where y = x, that means that x^{2} = x^{2}. That's fine and always true mod 2, but mod an odd prime it means that we have to have x^{2} = 0. The latter puts a significant restriction on what anticommutative structures are possible mod an odd prime that isn't present working mod 2. 
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The only place I already knew about where x^{2} = 0 is in a form of nonstandard analysis where d^{2} = 0, such that differentiation can be done algebraically. 
Aren't the integers mod 2 a pretty trivial arithmetic to be doing anything with, though?

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I deliberately left out a lot of detail. In algebraic topology we often work with rings graded on the integers (or more exotic things), and those rings are anticommutative in the sense that yx = (1)^{deg x • deg y} xy. So you get a negative sign if both x and y lie in odd grading, but if either lies in even grading the product is commutative. This means, for example, that if you want to define something like a polynomial algebra F_{p}[x] with x in a nonzero grading, if p is odd and you want x to have grading 1, you can only get a truncated algebra, with x^{2} = 0. On the other hand, if p = 2, you can have x in grading 1 and get a full polynomial algebra. 
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