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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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    Related, maybe even a duplicate: http://math.stackexchange.com/questions/111440/examples-of-apparent-patterns-that-eventually-fail2012-08-02
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    I don't think they're the same question. The one you linked is about large counterexamples, this is about small ones.2012-08-02
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    @SiliconCelery: *All* finite numbers are small, merely by the virtue that there are only finitely many smaller. For some people even countable sets are small and when talking about things like Woodin cardinals then pretty much every feasible set is nothing more than a tiny speck of insignificance.2012-08-02
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    $\mathbb R^n$ remains connected after the removal of a point, unless $n=1$. I'm sure I don't understand the question.2012-08-02
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    This is something like http://mathoverflow.net/questions/101463/properties-of-natural-numbers-such-that-there-is-a-very-large-largest-number-wi/101742#101742, e.g., there is no number $n$ whose every prefix is prime except when $n\le73939133$.2012-08-02
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    Quite a lot of the examples below have as special number/limit 4. I wonder if this is just a coincidence, or if some of those examples are related to each other.2012-08-03
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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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