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Motivated by How does the parity of a dimension affect its properties? I dare to ask the following question (with thanks to my colleague Vedran Dunjko): We happen to live in a world of prime dimensionality. Is that special? Are there properties of $\mathbb{R}^n$ that only apply to $n$ a prime integer?

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  • $\begingroup$ For one, various representation theory questions are much easier if the dimension is a prime number. No nontrivial imprimitive groups, etc etc. $\endgroup$ – Dima Pasechnik Oct 1 '18 at 11:20
  • $\begingroup$ Quite obviously, there seems to be some kind of irreducibility attached to real linear spaces of prime dimension : such a space can't be isomorphic to $ \mathbb{R}^{m}\otimes\mathbb{R}^{n} $ with both $ m $ and $ n $ greater than $ 1 $ . $\endgroup$ – Sylvain JULIEN Oct 1 '18 at 11:24
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    $\begingroup$ I can hardly wait for "does having a Fibonacci number of dimensions affect its properties?" $\endgroup$ – Gerry Myerson Oct 1 '18 at 12:41

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