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Let a-f be integers g.t. 2 with $a < b < c < d < e < f$. Let

$\ln def - \ln a b c = \alpha.$

Let $\{p_i\}$ be the set of prime factors (with repetitions) in a,b,c. Let $\{q_i\}$ be the set of prime factors in d,e,f. Then we know that

$ \ln def = \ln \prod q_i = \sum \ln q_i$ and so for $p_i$ and a,b,c.

Then $\sum \ln q_i - \sum \ln p_i = \alpha .$

It is true I think that $def - abc \gg \ln def - \ln abc.$ My question is, can we make any quantitative statements $ \alpha = f (\beta)$ about

$\sum q_i - \sum p_i = \beta$ based on our knowledge of $\alpha$? It's tempting to say that $\alpha < \beta$, for example, but I don't see how to prove it.

Thanks for any suggestions/answers.

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    The symbol you wanted there is $\gg$, produced by `\gg`.2012-07-22

1 Answers 1

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$17\lt19\lt23\lt25\lt27\lt32$ but $17+19+23\gt5+5+3+3+3+2+2+2+2+2$ so $\alpha\gt0$ but $\beta\lt0$.

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    Yes, am doing so.2012-07-23