Taking the canonical form of the fundamental theorem of arithmetic in the form: $$n=\prod_{j=1}^\infty p^{m_j}_j \qquad ;m_j\in \Bbb N_0$$ Does anybody know about a $\log n$ transform of this?

Note: also reference to other works are welcome


closed as not a real question by Gottfried Helms, Zev Chonoles, Davide Giraudo, Martin, Grigory M Jun 15 '13 at 9:38

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    $\begingroup$ What do you mean by a "$\log n$ transform" of the theorem? There is an internal direct sum decomposition of additive groups $\sum_{n\ge1}(\log n){\bf Z}=\bigoplus_p (\log p){\bf Z}$, if that's at all relevant. $\endgroup$ – anon Jun 4 '13 at 20:50
  • $\begingroup$ What about it? What is it that you want to know that you don't already know? $\endgroup$ – anon Jun 4 '13 at 21:24
  • $\begingroup$ I have never seen somewhere this form, although it seems so trivial. I was just wondering whether someone knows about this form elsewhere. $\endgroup$ – al-Hwarizmi Jun 4 '13 at 21:37
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    $\begingroup$ There is an answer on MSE somewhere (of Bill D) that proves $\{\log p\}$ are independent over the rationals using FToA. Otherwise I don't see any point in a text mentioning this form. $\endgroup$ – anon Jun 4 '13 at 22:14
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    $\begingroup$ No, it was this one. $\endgroup$ – anon Jun 5 '13 at 6:09

You don't clarify what "$\log$ transform" of FTA would look like. A trivial reformulation of FTA is that the log of any positive natural can be written uniquely as a $\bf N$-linear combination of logs of primes, but there is little aesthetic appeal in this formulation. Another algebraic version is this:

$$\log\left({\bf Q}^\times_{>0}\right)=\bigoplus_p (\log p){\bf Z}.$$

One application of this log perspective though is in exhibiting an infinite $\bf Q$-linearly independent set of real numbers, thereby proving that $\bf R$ is an infinite-dimensional $\bf Q$-vector space via arithmetic.

  • $\begingroup$ excellent. thanks. Do you have a reference for the latter possibly where I could read further? $\endgroup$ – al-Hwarizmi Jun 15 '13 at 10:50
  • $\begingroup$ @al-Hwarizmi What latter possibility are you referring to? The direct sum isomorphism in the middle of this answer should be a basic exercise if you know abelian group theory and requires no reference outside of such. Using logs to show $\bf R$ is an infinite-dimensional $\bf Q$-vector space is also a pretty basic exercise, and I already linked to a reference in the comments. $\endgroup$ – anon Jun 15 '13 at 15:41

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