# Does the primality of the number of dimensions affect its properties?

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?

• For one, various representation theory questions are much easier if the dimension is a prime number. No nontrivial imprimitive groups, etc etc. – Dima Pasechnik Oct 1 '18 at 11:20
• 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 $1$ . – Sylvain JULIEN Oct 1 '18 at 11:24
• I can hardly wait for "does having a Fibonacci number of dimensions affect its properties?" – Gerry Myerson Oct 1 '18 at 12:41