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In studying mathematics, I sometimes come across examples of general facts that hold for all $n$ greater than some small number. One that comes to mind is the Abel–Ruffini theorem, which states that there is no general algebraic solution for polynomials of degree $n$ except when $n \leq 4$.

It seems that there are many interesting examples of these special accidents of structure that occur when the objects in question are "small enough", and I'd be interested in seeing more of them.

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    @celtschk, it's also interesting to note that in many of the formulas for the laws of physics (at least in mechanics and electricity and magnetism, which is as far as I've gotten), the exponents and coefficients are rarely greater than 4...2012-08-04

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You might enjoy the article The Strong Law of Small Numbers by Richard K. Guy.

He has other publications in which he partly recycles the title.

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Another accident of small dimension: in all dimensions $\gt 4$, there are only three regular polytopes: the simplex, hypercube, and cross-polytope (the dual of the hypercube). In two dimensions there are infinitely many (all of the $n$-gons); in three dimensions you have two additional polyhedra, the dodecahedron and icosahedron; and in four dimensions there are three additional ones, the 120-cell and 600-cell (which are dual to each other) and the 24-cell (which is self-dual).

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    Thank you, this is a very nice example.2012-08-03
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The 26 sporadic groups is also an "accidental" occurrence. Since there is a finite number of them, after some large N, all the simple groups of order N or larger fit into a given category.

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    This is probably my favorite. Though it's perhaps strange to think of the [Monster](http://en.wikipedia.org/wiki/Monster_group) as a coincidence of small order. :)2012-08-03
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The outer automorphism group of $S_n$ is trivial, except when $n=6$.

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    Out(S2)=1 is still trivial, it just isn't equal to S2.2012-08-03
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The $n^\mathrm{th}$ cyclotomic polynomial is the minimal polynomial whose roots are the primitive $n^\mathrm{th}$ roots of unity, that is

$\Phi_n(X) = \prod_{ {0\leq j < n} \atop {\gcd(j,n)=1}}(X - e^{2\pi i j / n})$

The first few examples are: $\Phi_1(X) = X - 1 $ $\Phi_2(X) = X + 1 $ $\Phi_3(X) = X^2 + X + 1$

For all $n$ small enough, all the coefficients are $\pm 1$ or $0$. But if at least three odd primes divide $n$ (which requires at least $n\geq3\cdot5\cdot7 = 105$), other coefficients are possible.

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The alternating group $A_n$ is simple for $n\neq 4$.

This is related to the example given by the thread creator since it allows $S_4$ to be solvable, thus guaranteeing that all polynomials of degree $4$ or less are resolvable with only the field operations plus the extraction of roots.

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Deciding whether a graph is $2$-colorable has an obvious polynomial-time algorithm.

Deciding whether a graph is $k$-colorable for $k \geq 3$ is already NP-Complete!

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    There's a lot of structure in the 'booleanness' of two, in particular arguably the fact that $S_2$ is abelian (and so one can rearrange products of elements of $S_2^n$ arbitrarily), that tends to make all of these problems 'easy' for $n=2$.2014-05-25
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Similar to the answer about the $26$ sporadic groups and nontrivial $\mathrm{Out}(S_n)$ for $n=2,6$, there are a bunch of what are called exceptional isomorphisms that occur with low order/dimension. Basically, groups come in all sorts of special families (where there is a rule designating what groups are in the family and why) that are infinite, but these infinite families share a few finite cases between them.

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Orthogonal Latin squares exist for every order except 2 and 6. Euler conjectured from the small examples that they existed for any order not of the form $4k+2$, but he was mistaken.

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$\mathbb{R}^n$ has a unique smooth structure except when $n=4$. Furthermore, the cardinality of [diffeomorphism classes of] smooth manifolds that are homeomorphic but not diffeomorphic to $\mathbb{R}^4$ is $\mathfrak{c}=2^{\aleph_0}$.

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I would include in this list a discussion of $G(n)$ and $g(n)$ in the Waring Problem.

The point is that when it comes to representing integers as sums of powers of non-negative integers, it seems to happen that some (smallish) integers require more powers to represent them due to some some (presumably unknown) peculiarity of small integers.

For example, in the case of representing integers as sums of cubes, it has been proved that 9 cubes are sufficient, and some numbers require 9, so that $g(3)=9$.

On the other hand, calculations suggest that almost all numbers from some point onwards are sums of at most 4 cubes (so that $G(3)$ might be 4, but this is not proved), and it appears that it is an "accident of small $n$" that there are some (smallish) numbers which require more than 4 cubes.

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    $Y$ou are probably right - I had always assumed it was something to do with quirks of small numbers, but your explanation seems more likely.2012-08-03
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The ring of integers of $\mathbf{Q}[\sqrt d]$ is a principal ideal domain for $1 \leq d < 10$, but not for $d=10$.

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    Yes, you're right. I was thinking 2 was irreducible in that ring, since there's nothing of norm 2, but there are elements of norm $-2$, and they do the trick.2012-08-04
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Every integer is less then 100, except for $n>99$, where the pattern breaks.

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    I suppose I should have specified "nontrivial examples".2012-08-04
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The sequence of the maximal number of regions formed by drawing chords between all pairs of n points in arbitrary position on the border of a circle starts 1, 2, 4, 8, 16; and then the next term, of course, is... 31. (The canonical formula is $1+{n\choose2}+{n\choose4}$.) For more details, see http://oeis.org/A000127

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    It's fun to point out to people that the sequence contains 256; that provides extra "evidence" that it consists of powers of two!2012-08-24
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Let $n$ be fixed and let $x_1, x_2, \ldots x_n$ by any positive numbers, where the indices are taken mod $n$, so that $x_{n+i}$ is understood to be the same as $x_i$.

It is easy to show that $\sum_{i=1}^n \frac{x_i}{x_{i+1}} \ge n$

It is tempting to conjecture similarly:

$\sum_{i=1}^n \frac{x_i}{x_{i+1}+x_{i+2}} \ge \frac n2.$

And indeed, you would be hard-pressed to find a counterexample, since the conjecture is true…

if $n$ is one of: $\left\{\begin{array}{cccccc}1,&2,&3,&4,&5,&6,\\ 7,&8,&9,&10,&11,&12,\\ 13,&&15,&&17,\\ 19,&&21,&&23 \end{array}\right\} $ In all other cases, the conjecture is false.

(Shapiro's inequality)

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I wouldn't really call them accidents, but here are two simple related results from algebra:

There exists algebraic extensions of $\mathbb{R}$ of dimension $n$ only for $n=1$ and $n=2$.

Also it is not possible to construct an Division Ring over $\mathbb{R}$ of dimension $n$, excepting when $n=4$.

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On the congruence $(x-1)! \mod x$:

  • This is $0$ if $x>4$ is composite.
  • This is $1$ if $x$ is prime.
  • This is $2$ when $x=4$.

One can say that $x=4$ is here the exception.

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There are cyclic numbers for all bases $>4$, except for perfect squares, and $6$ (if you exclude leading zeros).

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Although there is an infinite family of generalized Petersen graphs with arbitrarily many vertices, there are exactly seven of these that are edge-transitive, the largest having 48 vertices.