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Is the set $\{\sin (p_{i})| p_{i} \,\,\mbox{is a prime number, for all}\,\,i\in \mathbb{N}\,\,\mbox{and}\,\, p_{i}\neq p_{j}\,\,\mbox{if}\,\,i\neq j\}$ is linearly independent when $\mathbb{R}$ is a vector space over $\mathbb{Q}$?

Thanks for any help.

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Nice Question, I haven't thought about this kind of things too. – user21436 Feb 1 '12 at 22:35
up vote 12 down vote accepted

I take the question to be whether $\{\sin p\mid p\hbox{ prime}\}$ is linearly independent over $\Bbb Q$. The answer is yes. Let $i:=\sqrt{-1}$. Since $\sin n = (2i)^{-1} (e^{in} - e^{-in})$, it suffices to prove, more generally, that $\{e^{in}\mid n\in{\Bbb Z}\}$ is linearly independent over $\Bbb Q$. This follows from the transcendence of $e^i$, which is proved by the Lindemann-Weierstrass theorem.

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