Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
Nice Question, I haven't thought about this kind of things too. –  user21436 Feb 1 '12 at 22:35

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.