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I would like to ask whether there is a proof of the following statement: Let $p$, $q$ be primes and $n$ positive integer coprime with $pq$. Then $\frac1{\log p}$, $\frac1{\log q}$ and $\frac1{\log n}$ are linearly independent over the rationals.

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A much stronger statement follows from Schanuel's conjecture (en.wikipedia.org/wiki/Schanuel's_conjecture), namely that the logarithms of the primes are algebraically independent. –  Qiaochu Yuan Apr 11 '12 at 23:14

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