I'm interested in very natural examples in math where something goes askew in a big way after some low integers. In topology, it's known that the n-dimensional sphere (the unit sphere in R^(n+1) cannot have more than one smooth structure if n = 1, 2, 3, 5, 6. Almost every sphere of dimension >= 7 does have more than 1. (The case n = 4 is unresolved.) As Adam pointed out, the symmetric and alternating groups S_n, A_n, have some strange things happening for low n. Like A_n is simple, except not for n = 4. The outer automorphism group Out(S_n) is trivial for all n *except* for n = 6 (there is an outer automorphism). What are the best examples of this kind of thing in various branches of math? I'm not so interested in things like "Every integer greater than 127401 is the sum of 17 9th powers." I just made that up, but you get the idea. —Dan