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I have rather a "toy" type of modelling-problem that appeared to me along a book I am writing on number theory. I would be outmost thankful for any concrete or inspirational answers, including comments, examples, ideas, references and more and more:

By applying natual $ln$ to both sides of the canonical form of the fundamental theorem of arithmetic, it is simply possible to express any natural number $n\in \Bbb N$ in a linear combination as sum of weighted primes $p\in \Bbb P$:

$$\ln n=\sum_i m_i \ln p_i$$

In many ares of physics $n\in \Bbb N$ is often used to mark up states, harmonics...

In a storm of thoughts: how could we interpret a transformation of all $n\in \Bbb N$ in primes $p\in \Bbb P$ in different araes of physics. The composites will give place to primes, for instance: prime quantum states instead of n-states... What are theoretical modelling conseuqences? Could we comprehend things different or better?

Look foward for your many inspirations...

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As you know, physicists often try to really calculate quantities. Indeed in physics you encounter sums of the form $\sum_n \ln n f(n)$, for example (where $f$ takes cares of the convergence). The problem here is that your $m_i$ and $p_i$ depend on $n$ (implicitly), so there is no simple way to input your formula in such a quantity. But maybe someone else has a more abstract idea. –  Vibert May 26 '13 at 9:14
    
$n=e^{\sum_i m_i \ln p_i}$ helps? –  al-Hwarizmi May 26 '13 at 9:17
    
That's not very different, right? ;) As an afterthought, in some cases mapping to primes can be useful, but in a strictly mathematical sense. Think of the Euler product for the zeta function or related identities. In some cases, a sum/product over primes might have better convergence properties than a sum over the integers. –  Vibert May 26 '13 at 9:32
    
Looking just at the formula, I thought this is gonn be about entropy ... –  Dilaton May 26 '13 at 9:47
    
But what if one could show that for instance a wave can be completely factorised in fundamental modes of prime frequencies $f(p)$? –  al-Hwarizmi May 26 '13 at 10:53
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