I’m struggling to prove that the set $\{\ln(p): \text{$p$ is prime}\}$ is a linearly independent system in the Rational vector space of the real numbers.

Any help is greatly appreciated.


marked as duplicate by Stefan Mesken, Martin Sleziak, Crostul, Siong Thye Goh, David K Dec 11 '17 at 19:21

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    $\begingroup$ hint: reduce everything to same denominator to get an integer relation. transform aln(p)=ln(p^a) and ln(x)+ln(y)=ln(xy). take exponential and finally use Euclid's lemma. Can you redact this in your question ? $\endgroup$ – zwim Dec 11 '17 at 17:49

Hint: Use contradiction. suppose it is not linearly independent. $$\ln(p_i) = \alpha_1\ln(p_j) + \alpha_2\ln(p_k) = \ln(p_j^{\alpha_1}p_k^{\alpha_2 })$$

So, $p_i = p_j^{\alpha_1}p_k^{\alpha_2 }$. As all $p_i$, $p_j$, and $p_k$ are prime, it can not be possible. To be more precise, you should use induction to show this.


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