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I am currently reading a paper which takes for granted the following geometric fact: if $\mathbb{R}P^n$ can be immersed in $\mathbb{R}^{n+1}$ then for some $k$, $n=2^k-1$ or $n=2^k-2$. My initial thought was that this has something to do with Stiefel-Whitney numbers, but I can't see how that would work. Thoughts?

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