Is the set $S:=\{\ln p \mid p\ \text{is prime}\}$ is algebraically independent over $\Bbb Q$? i.e. if $t_1 \lt t_2 \lt \cdots \lt t_n \in S$, then $f(t_1,t_2,\cdots ,t_n) \neq 0$ for all nonzero polynomials $f \in \Bbb Q [X_1,X_2, \cdots,X_n]$ ?
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2Looking at the comments to this question and at wikipedia, it appears that not even the existence of two prime numbers $p_1,p_2$ such that $\ln p_1$ and $\ln p_2$ are algebraically independent has yet been proved. – Jul 10 '17 at 15:54
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@G. Sassatelli Now I've read the wiki's description. The Japanese wiki does not refer to this... Thank you for your comment! – p_i Jul 10 '17 at 17:50