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If someone gives me a list of numbers and says they are entirely and completely random, how I can I verify this?

EDIT: Let's suppose that a well-known string theorist told me he can produce a list of numbers that is truly, genuinely random. If I can't prove that they're really random, then who am I supposed to believe: him, or the mathematicians?

(Don't hurt me; I'm just trying to learn.)

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You can’t. For any finite sequence of numbers $\langle x_1,\dots,x_n\rangle$ there is a polynomial $p$ of degree $n-1$ such that $p(k)=x_k$ for $k=1,\dots,n$. –  Brian M. Scott Dec 8 '12 at 7:31
    
    
Wait, it's not quite a duplicate. I need to get some answers first and then I'll add an edit to my post. But I don't want to influence the answers I get with additional information. –  Nick Dec 8 '12 at 7:34
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"I don't want to influence the answers I get with additional information." That sounds like a really bad idea. Why would anyone want to answer if (a) they don't know what you're looking for that's different from the already existing previous questions and answers, and (b) after they've put in all the effort to write a nice answer, you might change the question rendering their answer irrelevant? Tell us what your question really is, please. –  Rahul Dec 8 '12 at 7:47
    
See my edit above. –  Nick Dec 8 '12 at 7:52
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