From Baker's theorem it follows that a linear combination of natural logarithms of prime numbers with non-zero algebraic coefficients can never be zero.

Has it been proved that the set of all natural logarithms of prime numbers is algebraically independent over $\mathbb Q$?

In other words, can we be sure that expressions like the following are never exactly zero? $$\frac{347}{75}\,\ln^22\cdot\ln3-\frac{173}{100}\,\ln2\cdot\ln^23+\frac{179}{180}\,\ln^32-\ln^33$$

  • 2
    $\begingroup$ As the article you link to states, next to nothing is known about this sort of question: "so far it remains to be proved that there even exist two algebraic numbers whose logarithms are algebraically independent." $\endgroup$ – lulu Jul 27 '15 at 17:06
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    $\begingroup$ As far as I know this is an open problem, but like many open problems of this form it follows from Schanuel's conjecture (en.wikipedia.org/wiki/Schanuel%27s_conjecture). $\endgroup$ – Qiaochu Yuan Jul 27 '15 at 17:07

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