# covering map from spheres to projective spaces and the associated vector bundle

Let $S^n$ be the $n$-sphere and consider a $2$-sheeted covering $$S^n\longrightarrow\mathbb{R}P^n.$$ We have an associated vector bundle $$\xi: \mathbb{R}^2\longrightarrow S^n\times_{\mathbb{Z}/2}\mathbb{R}^2\longrightarrow \mathbb{R}P^n$$ where the nontrivial element of $\mathbb{Z}/2$ acts on $\mathbb{R}^2$ by reversing the order of coordinates.

Question: What is the smallest positive integer $k$ such that the Whitney sum $\xi^{\oplus k}$ is stable equivalent to a trivial bundle?

Theorem 7.4 of J. F. Adams, Vector fields on spheres, Ann. of Math. 75 (1962), 603–632 says that $$\tilde{KO}({\mathbb R} P^n)=\mathbb Z\,/\,2^{\phi(n)},$$ generated by $\xi-1$, where $\phi(n)$ is the number of integers $s$ such that $0 < s\le n$ and $s$ is congruent to 0,1, 2 or 4 modulo 8, and $\xi$ is the canonical line bundle over $\mathbb RP^n$ given as $S^n\times_{\mathbb Z/2} \mathbb R\to \mathbb RP^n$, where $\mathbb Z/2$ acts on $\mathbb R$ by multiplication with $-1$.

Since your vector bundle is the direct sum of $\xi$ and the trivial 1-dimensional bundle, the answer to your question is $k=2^{\phi(n)}$.