I have read Gauge Theory on Asymptotically Periodic 4-Manifolds by Clifford Henry Taubes where an uncountable family of diffeomorphism classes of oriented 4-manifolds which are homeomoprphic to $\mathbb{R}^4$ is constructed.
Furthermore, I know that for 1-, 2-, and 3-manifolds homeomorphic manifolds are already diffeomorphic. Thus, all $\mathbb{R}^n$ with n < 4 have a unique smooth structure up to diffeomorphism. The same holdes for n > 4 as many textbooks and wikipedia claim without proof.
Unfortunately, I have not any idea how to prove that $\mathbb{R}^n$ for every n > 4 has a unique smooth structure up to diffeomorphism. Does anybody know the proof or a paper where it has been proved?