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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.

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1  
No $\ell_p$ space is a Hilbert space (except when $p=2$)? – Clement C. 1 hour ago
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$\frac{x}{x}=1$ for all real numbers except $x=0$. – Joel Reyes Noche 1 hour ago
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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 1 hour ago
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There are infinite kind of proofs that are true for all cases but one. Making a list is absurd. – Masacroso 1 hour ago
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If $x.y$ are positive integers then $x^2+2\ne y^3$ except when $ x=5$ and $y=3.$ (Pierre de Fermat) – user254665 46 mins ago
up vote 13 down vote accepted

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:

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

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

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Would absolutely love a short accessible introduction to the structures/theorems underlying this, besides the links provided. If anyone has one. – D. W. 24 mins ago
up vote 4 down vote

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 $S_n$

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up vote 3 down vote

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

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

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

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up vote 0 down vote

$\varphi(n)$ is always even , for all $n\in N $ except n=2.

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up vote 0 down vote

All primes are odd except for 2.

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Thank you all! I am currently looking these things up and I am fascinated.

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Please don't add "thank you" as an answer. Instead, accept the answer that you found most helpful. - From Review – suomynonA 27 mins ago

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