What are some interesting sole exceptions or counterexamples? [duplicate]

Solution 1:

The smooth structure on $\mathbb{R}^n$ is unique up to diffeomorphism, except if $n = 4$.

Solution 2:

If the $n$th Fibonacci number is prime then $n$ is prime, except that $F_n=3$ when $n=4$.

Solution 3:

A generalization of the Four Color Theorem says that the chromatic number of a closed surface with Euler characteristic $\chi$ (the number of colors needed to color any map on the surface) is bounded above (sharply) by $$\left\lfloor \frac{7 + \sqrt{49 - 24 \chi}}{2} \right\rfloor,$$ except for the Klein bottle, which has Euler Characteristic zero, so that the above formula gives a count of $7$ colors, but which only requires $6$. (See Ringel, G. and Youngs, J. W. T. Solution of the Heawood Map-Coloring Problem. Proc. Nat. Acad. Sci. USA 60, 438-445, 1968.)

(This is borrowed from this answer to the question JimmyK4542 references in the comments above.)

Solution 4:

The group of units of $\mathbb Z/p^n \mathbb Z$ is cyclic for every prime power $p^n$, except when $p=2$; then it's cyclic only for $n=1,2$.

Solution 5:

The duals of $L^p$ spaces: $$1\le p<\infty,\ \frac1p+\frac1q = 1\implies (L^p)^*=L^q,$$ but $$(L^\infty)^*\ne L^1.$$