I know there are several big list questions out there (e.g. Patterns that break down at certain numbers) that touch on classifications of mathematical structures where certain numbers don't fit in, but I'd like to find more explanations on why these numbers are special.
For instance, all automorphisms of symmetric groups are conjugacies, except for $n=6$, where the conjugacies form a subgroup of index two. The reason 6 is special is that the number of partitions of six distinct objects into two sets of three is the same as the number of partitions into three sets of two.
As another example, 3-manifold groups are special because in higher dimensions any group can be a manifold group (by Dehn surgery), but we can still embed all graphs into all 3-manifolds (which makes them more interesting than 2-manifolds).
Some special numbers I'd like to see explanations for are:
- Fermat's last theorem: where does the proof break down for $n=2$? (I know there are obvious counterexamples, but why do the techniques used require $n>2$)?
- Exotic smooth structures in dimension 4.
- The finitely many quadratic extensions of the rationals that are not Euclidean or ufd's, etc.