Is there a proof that is true for all cases except for exactly one case?

Question

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.

Answer 1

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

Answer 2

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 $S_n$

Answer 3

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

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

Answer 4

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 $$\gamma(\chi) = \left\lfloor \frac{7 + \sqrt{49 - 24\chi}}{2} \right\rfloor.$$

Answer 5

All primes are odd except for 2.

Answer 6

For all real numbers $x$,

$$x \neq 1$$

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

Answer 7

• 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.

Answer 8

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).

Answer 9

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.

Answer 10

Any polygon with $n$ sides can be concave.

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

Answer 11

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: $\pi_1(S^1) = \Bbb Z$.