I found this problem on Math.SE:

Prove that $\log_35+\log_25$ is irrational.


I labored on it for a few days, and couldn't find an algebraic solution- I'm not even sure if such a solution exists. All I was able to do was prove that both components were irrational by themselves (as opposed to their sum). I am wondering if anyone has seen this problem before, and/or if anyone knows a solution. If so, I could really use a hint.

So far, using the Fundamental Theorem of Arithmetic (i.e., all integers have a unique prime factorization) hasn't helped me the way one would use it to show that the individual components are irrational.

Thank you in advance.

  • $\begingroup$ To make this question more self contained, I would recommend giving the number itself. $\endgroup$ – Joonas Ilmavirta Oct 27 '14 at 20:43
  • $\begingroup$ Will do. Editing it now. $\endgroup$ – daOnlyBG Oct 27 '14 at 20:43
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    $\begingroup$ One would expect this sum to be transcendental, but even proving irrationality of the sum of two given transcendental numbers tends to be hard. $\endgroup$ – Stefan Kohl Oct 27 '14 at 21:19
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    $\begingroup$ Incidentally, Schanuel's conjecture would imply that this number is transcendental. $\endgroup$ – Henry Cohn Oct 27 '14 at 21:23
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    $\begingroup$ This won't help much, but you can rewrite the number as $(\log5\log6)/(\log2\log3)$. $\endgroup$ – Gerry Myerson Oct 28 '14 at 5:40

To resonate with Henry Cohn's comment, Schanuel's conjecture implies that the natural logarithms of the primes are algebraically independent over $\mathbb{Q}$. In particular, the statement in the original post is probably true, but proving it might be out of reach at the moment.


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