# Is there a proof that is true for all cases except for exactly one case? [closed]

I was curious if there were any such proofs which state that a thing is true always EXCEPT for exactly one instance. As in, for some reason, there is only one instance where the proof is false, but it is true for all other objects. I understand that if it is not true in that one case that it is not necessarily a proof, I was just wondering if there were any "proof-like things" of this form.

• Commented Sep 30, 2016 at 2:55
• $\frac{x}{x}=1$ for all real numbers except $x=0$.
– JRN
Commented Sep 30, 2016 at 3:04
• All primes are odd except for $2$. That's not an entirely serious answer, of course, but just to point that there is a certain ambiguity in how you define except for exactly one case. Any theorem or proof that assumes p an odd prime (of which there are many) could technically be restated as assuming p prime, with the exception of 2.
– dxiv
Commented Sep 30, 2016 at 3:05
• There are infinite kind of proofs that are true for all cases but one. Making a list is absurd. Commented Sep 30, 2016 at 3:31
• @Semiclassical, did you mean differentiable? Commented Sep 30, 2016 at 7:34

Here's a famous one: $\mathbb{R}^n$ has a single differentiable structure (up to diffeo) except for $n=4$, in which case it has uncountably many.

These posts may be of interest:

https://mathoverflow.net/questions/16035/a-reference-for-smooth-structures-on-rn

https://mathoverflow.net/questions/24930/differentiable-structures-on-r3

• Would absolutely love a short accessible introduction to the structures/theorems underlying this, besides the links provided. If anyone has one. Commented Sep 30, 2016 at 4:19

For all positive integers $n$, the symmetric group $S_n$ has a trivial outer automorphism group, except for $S_6$, which has $2$ elements.

Related: Outer Automorphisms of $S_n$

For all real numbers $x$,

$$x \neq 1$$

This is true for all numbers except $1$, your single exception.

• This theorem can be extended to complex numbers as well :) Commented Sep 30, 2016 at 11:11
• Wait what about x = 0.999999…?! Just kidding Commented Sep 30, 2016 at 11:49

The Heawood conjecture as it applies to the Euler characteristic, is an example of a theorem that is true except in exactly one case. In the case where $\chi = 0$ for the Klein bottle, the minimum number of colors needed to color all graphs drawn on this surface is $6$, not $7$ as indicated by the formula $$\gamma(\chi) = \left\lfloor \frac{7 + \sqrt{49 - 24\chi}}{2} \right\rfloor.$$

For each prime $p\neq 2$ the following holds:

The multiplicative group of $\mathbb Z_{p^s}$ is cyclic for all $s\geq 1$

• Aren't there a lot of theorems which only apply to "odd primes?" Commented Sep 30, 2016 at 6:38
• Yes. ${}{}{}{}{}$ Commented Sep 30, 2016 at 12:56

All primes are odd except for 2.

• That is literally my comment which I posted earlier under the original question. You may want to add the rest of my comment, too, since it is part of the point being made.
– dxiv
Commented Sep 30, 2016 at 5:52
• Sorry. I was on mobile when I posted this and did not see that. Commented Sep 30, 2016 at 12:08

Disclaimer: This is only a comment. But a comment to all the answers. So I am posting it as an answer so that all would see this.

The question clearly asks for PROOF that is valid for all except one case, and does not ask for STATEMENTS (theorems).

So it is reasonable to assume that what is expected is a theorem that is true
(for all positive integers, all manifolds, all groups etc) but a proof that is valid except in one case.

KitCarpson should clarify.

Any polygon with $n$ sides can be concave.

This is true for all $n$ except $n=3$, i.e. all triangles are convex.

Here is yet another example, the Brooks' theorem: all vertices of each connected undirected graph where the degree of each vertex doesn't exceed some $d$ can be colored in $d$ colors so that no two incident vertices will have the same color (except for complete graph which has $d + 1$ vertices).

• Did you go to IMC? Commented Sep 30, 2016 at 12:59
• @JorgeFernándezHidalgo Yes Commented Sep 30, 2016 at 13:18
• I started using guake because I saw your screencast once, it's a pretty good setup :) Commented Sep 30, 2016 at 13:56

Trivial example: taking $z = e^{i\theta}$ can be proved easily that for $n\in\Bbb Z$: $$\int_{|z|=1}z^ndz = 0\iff n\ne -1.$$

Nontrivial example: the Hartogs's extension theorem:

Let $f$ be a holomorphic function on a set $G\setminus K$, where $G$ is an open subset of $\Bbb C^n$ $(n\ge 2)$ and $K$ is a compact subset of G. If the relative complement $G\setminus K$ is connected, then $f$ can be extended to a unique holomorphic function on $G$.

Another nontrivial example: all the fundamental groups of the spheres are trivial except in dimension 1: $\pi_1(S^1) = \Bbb Z$.

• Each object you can touch in your body is not your tongue, but it fails when the object you touch is your tongue.

• Each person you look in this planet is not your mother, but it fails when you look at your mother.

• Each person in this question is trying to showoff with advanced math concepts to answer a trivial question like this and is not called Voyska, but it fails for me.