# Oddities of evenness

Being initially a little bit perplexed by the observation that the possibility of calculating vertex potentials $$\lbrace\pi_1,\dots,\pi_n\rbrace$$ for weighted cycle graphs $$C_n,\,2\lt n$$ such that the potentials $$\pi_u,\pi_v$$ of the vertices $$u,v$$ that are adjacent to the edge $$(u,v)$$ sum exactly to that edge's weight $$|(u,v)|$$, i.e. $$\pi_u+\pi_v=|(u,v)|,\ \forall (u,v)\in C_n$$, depends on the parity of $$n$$, i.e. a solution exist for $$n=2k+1$$ but not for $$n=2k$$.

Therefore I would like to ask the

Question:

what are other non-trivial examples of mathematics where the parity of an integral parameter makes a crucial difference?

I'm especially hoping for examples that are not classical results, but don't rule them out.
Previously unknown results from personal research would of course be the most preferable answers.

• A classical result (I know you don't want it, but I'm going to mention it anyway) is that a polynomial with real coefficients is guaranteed to have a real zero if its degree is odd, no such guarantee if its degree is even. Here's another: $\int x^ne^{-x^2}\,dx$ is elementary if and only if $n$ is odd. May 30 at 1:17
• Not sure what kinds of answers are expected. Parity is ubiquitous in combinatorics and graph theory, e.g. many counting formulas have a parity-dependent term. But I guess most of these would go to the "trivial" bin. May 30 at 7:13
• Does Feit-Thompson qualify? May 30 at 20:24
• What I want is juicy examples where what matters is the value of an integer mod 3. May 31 at 21:56
• @JohnBaez Not sure how juicy this is, but here is a restriction estimate of Bourgain and Guth that depends on the dimension mod 3: mathscinet.ams.org/mathscinet-getitem?mr=2860188 . (The paper also establishes an oscillatory integral estimate is new only in even dimensions, so technically also answers the OP question.) May 31 at 22:17

The hairy ball theorem is only valid for even-dimensional spheres (or odd-dimensional ambient Euclidean spaces).

Similarly, the strong Huygens principle is only valid in odd-dimensional physical spaces (or even-dimensional spacetimes).

Even-dimensional manifolds can admit symplectic structures, whilst odd-dimensional manifolds can admit contact structures.

• very nice examples; the mere existence of examples that are so diverse and not at all related to number theory makes me wonder if oddness or evenness are just the tip of a mathematical "iceberg" May 30 at 18:46

1. A polynomial with real coefficients is guaranteed to have a real zero if its degree is odd, and there is no guarantee of a real zero if its degree is even.

2. For integer $$n\ge0$$, $$\int x^ne^{-x^2}\,dx$$ is elementary if and only if $$n$$ is odd.

3. For integer $$n\ge2$$, $$\zeta(n)$$ is known for $$n$$ even, in the sense that it is a rational multiple of $$\pi^n$$, with a simple expression for the rational in terms of Bernoulli numbers. It follows that it is transcendental for even $$n$$. Much less is known about $$\zeta(n)$$ for odd $$n$$.

I think to fail to mention the famous Feit-Thompson theorem would be a crime!

Theorem: (Feit-Thompson) Let $$G$$ be a finite nonabelian simple group. Then $$G$$ has an even number of elements.

(The finite abelian simple groups are just the cyclic ones of prime order.)

The only proof we know is impressive, but not very enlightening.

The proof considers a minimal counterexample $$G$$: a nonabelian, simple finite group with an odd number of elements, all of whose proper subgroups are solvable. A lot of work is done to pin down the structure of $$G$$, until a final contradiction is derived, showing that $$G$$ cannot exist.

The fact that $$G$$ has an odd number of elements is used in a number of different ways (some of which are listed on Wikipedia).

The dream...

is to find a new perspective on finite groups that provides a more conceptual reason why nonabelian finite simple groups must have even order.

Sometimes I fantasise about developing such a perspective and arriving at a short equation cotaining the term $$(-1)^{|G|}$$, and the equation somehow determines the solvability of the group $$G$$. That's just a vague fantasy, though.

• It would be strange if the use of oddness was not crucial! May 31 at 17:06
• Sorry I wasn't clear! I meant to emphasise that the existing proof doesn't reveal any single reason why the oddness of its order "causes" a group to be solvable. I'll edit my answer. May 31 at 19:45

Perhaps too simple for what you are asking, but:

A proper edge-coloring of $$K_n$$ (the complete graph of $$n$$ vertices) requires $$n-1$$ colors if $$n$$ is even, but $$n$$ colors if $$n$$ is odd (and $$n>1$$).

• Well, for that matter, what about a usual coloring of the cycle graph... May 30 at 12:59
• Indeed! But I would argue that in the cycle, dependence on parity is almost "obvious", while in $K_n$ it is less obvious. Somehow one (or at least I) would expect $K_n$ to be "so overwhelmingly symmetric" that parity would not matter. May 31 at 10:22

Let $$\gamma :[0,1] \mapsto \mathbb{R}^{n}$$ be a curve such that all the leading principal minors of the matrix $$(\gamma^{(1)}(t), \ldots, \gamma^{(n)}(t))$$ are nonvanishing as $$t \in [0,1]$$.

Theorem: the minimal number $$k$$ points required to represent any interior point of $$\mathrm{Conv}(\gamma([0,1]))$$ as a convex combination of $$k$$ points of $$\gamma([0,1])$$ equals $$\lfloor \frac{n}{2}\rfloor+1$$.

Moreover, if $$n\geq 3$$ is even then there are only two possible corresponding convex combinations with $$\lfloor \frac{n}{2}\rfloor+1$$ points (one containing $$\gamma(0)$$, and the other one $$\gamma(1)$$). And if $$n$$ is odd then there is the unique corresponding convex combination with $$\lfloor \frac{n}{2}\rfloor+1$$ points (and it never contains any of these endpoints $$\gamma(0)$$ and $$\gamma(1)$$).

Another example: Closed strictly convex curves exists only in even dimensions. Recall that $$\beta :[0,1] \mapsto \mathbb{R}^{n}$$ is called strictly convex curve if it has at most $$n$$ common points with any hyperplane. Schoenberg proved an isoperimetric inequality for such curves "An isoperimetric inequality for closed curves convex in even-dimensional euclidean spaces"

The special orthogonal group $$SO_n$$ behaves quite differently depending on whether $$n$$ is even or odd. In the Cartan–Killing classification, the odd case is type $$B$$ and the even case is type $$D$$. The Dynkin diagrams, root systems, representation theory, etc., in types $$B$$ and $$D$$ are qualitatively different.

Finding the shortest odd-length directed cycle in a directed graph is a straightforward algorithmic problem. On the other hand, finding the shortest even-length directed cycle in a directed graph is an astonishingly subtle task, that was only very recently shown to be tractable. See The shortest even cycle problem is tractable, by Andreas Björklund, Thore Husfeldt, and Petteri Kaski.

There are many examples where parity matters a lot. Two classical examples from number theory:

In a different field, Sharkovskii ordering tells us that even periods are much more common among the set of real continuous functions.

In this preprint we find this:

Wave propagation in odd-dimensional space is fundamentally different from wave propagation in even-dimensional space. When the space dimension $$\mathrm D$$ is odd and $$\mathrm{D} > 1,$$ waves obey Huygens’ principle [1, 2]; that is, waves created by an instantaneous point source at $$t = 0$$ (e.g., a light pulse) take the form of an expanding bubble. After the wavefront passes by, the medium instantly returns to quiescence. An observer sees blackness until the wave arrives, sees an instantaneous flash as the wave passes by, and immediately afterward sees blackness again [see Fig. 1(a)]. Such a wave propagates on the surface of the light cone. In contrast, in even-dimensional space an instantaneous point source gives rise to a wave that develops a tail. An observer sees blackness until the wave arrives and then sees a flash. However, the medium does not immediately return to quiescence; rather, the wave amplitude decays to $$0$$ like $$t^{-\alpha},$$ where $$\alpha > 0$$ depends on $$\mathrm D.$$

• Already posted by T. Tao. Jun 1 at 18:26
• The quote is more informative than the link I currently have, though. Perhaps a merge? Jun 1 at 18:44
• @Terry: You can't merge answers, but this is why I've made the question a CW. As it's a list of examples, it's easier for everyone to edit and expand answers and there are no quibbles about reputation. Jun 1 at 23:15

The biggest little polygon is a regular polygon if the number of sides is odd, but is an unexpectedly interesting shape when the number of sides is even (and at least 6).

The $$L$$-function $$L(A,s)$$ of an abelian variety $$A/F$$ over a number field is conjectured to satisfy a functional equation relating $$s$$ and $$2-s$$ (this is known in many cases if $$F$$ is totally real). At $$s=1$$, so the center of the functional equation, the $$\epsilon$$ factor (of Serre, Deligne, Langlands) is just $$(-1)^r$$ for some integer $$r$$. The parity of $$r$$ has a tremendous impact on the arithmetic of $$A$$.

To begin with, if $$r$$ is odd, the $$L$$-function must vanish at $$s=1$$ so $$A$$ must have a non-torsion $$F$$-rational point according to the conjecture of Birch and Swinnerton-Dyer.

Bott periodicity implies that the (unstable) homotopy groups of the (infinite) classical groups are periodic. In particular, $$\pi_k(U)=\pi_{k+2}(U)$$ where $$U$$ is the infinite unitary group. This example is a little bit cheating because normally Bott periodicity is stated modulo 8 rather than modulo 2.

A very simple concept from geometry: Given a number of points with no three collinear, that are the midpoints of the sides of a polygon in rotational order, a unique solution generally exists for the polygon if the number of points is odd. With an even number of points, however, either the polygon fails to exist or is nonunique.

The claim is proven via a descent argument, which is demonstrated in the picture below for the pentagon having midpoints $$A,B,C,D,E$$. The pentagon is divided into a quadrilateral with three midpoints $$C,D,E$$ and a triangle itself midpoints $$A,B$$ which share a diagonal. The quadrilateral must have its fourth midpoint at $$F$$ which is then the third midpoint of the triangle. Assuming $$F$$ is not collinear with $$A$$ and $$E$$, The triangle is then found by drawing a line through $$A$$ parallel to $$E\overline{EF}$$ and cyclic permutations. Thus gives vertices $$G,H,K$$ of the pentagon. Then the collinearity and distance-doubling criteria give the rremainingpentagonal vertices $$I,J$$.

Thus with the parallelogram construction to determine $$F$$ the pentagonal problem is reduced to the simpler triangular one. In a similar way any set of $$n$$ points is reduced to $$n-2$$ points. Thus we can reduce all odd cases to a triangle (if the vertices of this triangle are not collinear), thus a unique solution; butcall even cases reduce to four points for which the solution exists only if the final four points are vertices of a parallelogram and then is nonunique.

There are a lot of results that are more difficult, or at least different, in characteristic 2 compared to odd characteristic. See for example the math.SE question, What's so special about characteristic 2? One example mentioned there is that the classification of simple Lie algebras over an algebraically closed field is seemingly intractable in characteristic 2.

This is an extension of Timothy Chow's answer about $$SO_n$$ behaving differently depending on the parity of $$n$$. Even when $$n=2m$$ (i.e., type $$D_m$$), the groups behave differently depending on the parity of $$m$$.

Since the center of $$SO_{2m}$$ is order $$2$$, we know the center of $$Spin_{2m}$$ is order $$4$$. But is it $$\mu_2^2$$ or $$\mu_4$$? The answer is: if $$m$$ is even, its center is $$\mu_2^2$$, while if $$m$$ is odd, its center is $$\mu_4$$.

Another example: $$O_{2m}$$ has nilpotent orbits (i.e., orbits of $$O_{2m}$$ acting on nilpotent elements in the Lie algebra $$\mathfrak{so}_{2m}$$) parameterized by partitions of $$2m$$ with an even number of even parts. In $$SO_{2m}$$ most nilpotents remain a single orbit, but those $$O_{2m}$$-orbits corresponding to partitions with only even parts becomes two $$SO_{2m}$$-orbits. Such phenomenon only occurs when $$m$$ is even.

A finite dimensional connected serial algebra (=Nakayama algebra) has finite global dimension(which is given as the supremum of the projective dimensions of all simple modules) if and only if there is a simple module of even projective dimension. This is a result by Dag Oskar Madsen.