Two Kinds of Odd

Actually I'd like to propose new names for all the congruence classes mod 4, as shown in this table.

	mod 4	base 2
eveve	0	00	…	−8	−4	0	4	8	12	…
evodd	1	01		−7	−3	1	5	9	13
odeve	2	10		−6	−2	2	6	10	14
ododd	3	11		−5	−1	3	7	11	15

I hope you can see the pattern there. Given a number, the congruence class mod 4 is determined by the last two digits of the number in base 2, and if you take those last two digits and read “0” and “1” as “ev” and “od”, you get the new names I'm proposing.

So, by the way, there are also two kinds of even numbers, “eveve” and “odeve”, but I won't have much to say about them. I dropped the final “n” from “even” so that all the names would have two syllables and five letters.

There is one tiny point that's counterintuitive. The negative of an even number is even, and the negative of an odd number is odd, right? Well, eveve and odeve numbers behave the same way, but evodd and ododd numbers get swapped. For example, 5 is evodd, but −5 is ododd, as you can see from the right-hand side of the table.

Now let's review some of those number-theoretic properties.

In Sums of Squares I explained how to calculate the number of ways that a number can be written as a sum of two squares, but somehow it didn't occur to me to mention this interesting corollary: an evodd number can always be written as a sum of squares, while an ododd number cannot. Without getting into the details, it's also generally true that the more powers of evodd primes a number contains, the more ways it can be written as a sum of squares.

If you're familiar with Gaussian integers, here's a related fact. Because (a+bi)(a−bi) = a²+b², a number can be written as a sum of squares a²+b² if and only if it can be written as a product (a+bi)(a−bi). Consequently, evodd primes can be factored over the Gaussian integers, while ododd primes cannot. In other words, ododd primes are also Gaussian primes!

There are some similar results for differences of squares (and differences of intermediates), but I'll let you read about those for yourself in Differences.

If a number has a square root mod m, we say that it's a quadratic residue mod m. For example,

4² = 16 = 17−1 ≡ −1 mod 17,

so −1 is a quadratic residue mod 17. Now, suppose p is an odd prime. It turns out that −1 is a quadratic residue mod p exactly when p is evodd. I explained why in Other Bases. The law of quadratic reciprocity also depends on whether the two primes are evodd or ododd.

Quadratic residues and the two kinds of odd numbers also show up in the last part of a long essay I wrote about repeat lengths of decimal expansions of fractions. The most interesting point is that the set of evodd numbers is closed under multiplication, while the set of ododd numbers isn't.

I'd need a lot more experience to have the same level of intuition for evodd and ododd numbers that I have for even and odd ones, but for what it's worth, the general sense I get is that just as odd numbers are troublesome because they don't divide evenly into two parts, ododd numbers are even more troublesome. For example, they don't decompose evenly into two squares.

Finally, I'd like to point out that the same system can be used to generate names for the congruence classes mod 8, mod 16, and so on. I don't think those classes are useful enough to need names, but you never know. I'd also like to point out the slight connection to octal and hexadecimal.

I do know one fact that involves the congruence classes mod 8, and here it is: 2 is a quadratic residue modulo an odd prime p if and only if p is congruent to ±1 mod 8 … in other words, if p is evevodd or odododd.

Here's a connection I should have made long ago. In The Structure of U_n, I said that for k ≥ 3, U_2^k = Z₂ × Z_2^k−2. Well, in that decomposition, there are two choices for each of the subgroups Z₂ and Z_2^k−2, and the natural choice is, take Z₂ to be {1,−1} and Z_2^k−2 to be … the evodd numbers!

The embarrassing thing is, if you follow the “last part” link above, you'll find the same thing in different words.

Second, such numbers—numbers that are congruent to 1 mod 4—form a cyclic subgroup of order 2^k−2. (You can use the first fact to prove the second … any number that ends in 101 has order 2^k−2 and so is a generator.) Third, you can get to the rest of the group by negating.

Also note that the number 5 (101) is a generator of Z_2^k−2 for every k. That's Theorem 3.9.2 in Introduction to Number Theory.