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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

al-Hwarizmi
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    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. – anon Jun 04 '13 at 20:50
  • What about it? What is it that you want to know that you don't already know? – anon Jun 04 '13 at 21:24
  • I have never seen somewhere this form, although it seems so trivial. I was just wondering whether someone knows about this form elsewhere. – al-Hwarizmi Jun 04 '13 at 21:37
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    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. – anon Jun 04 '13 at 22:14
  • wasnt that post on the squares "The square roots of the primes are linearly independent over the field of rationals"? – al-Hwarizmi Jun 05 '13 at 05:59
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    No, it was this one. – anon Jun 05 '13 at 06:09
  • This is great Help, anon. Many thanks. – al-Hwarizmi Jun 05 '13 at 06:34
  • @anon Please consider converting your comment into an answer, so that this question gets removed from the unanswered tab. If you do so, it is helpful to post it to this chat room to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see here, here or here. – Julian Kuelshammer Jun 15 '13 at 09:35
  • @JulianKuelshammer I converted my comments into an answer two hours before your comment asking me to; is yours a boilerplate comment? At any rate, thanks for the links. – anon Jun 15 '13 at 15:46
  • @anon Sorry then. I don't know why I didn't see it, maybe a bug, or I'm just blind. My comment is from the "list of comment templates" in meta. It was made by Lord_Farin and myself. – Julian Kuelshammer Jun 15 '13 at 15:51

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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.

anon
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  • excellent. thanks. Do you have a reference for the latter possibly where I could read further? – al-Hwarizmi Jun 15 '13 at 10:50
  • @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. – anon Jun 15 '13 at 15:41