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Introduction: The question that I really want to ask is, “Are there any well-known visual representations for the integers $\ge 2$, with perhaps one of them even being regarded as canonical?” However, since concrete questions are almost always more interesting/memorable/concise than abstract questions, I gave the title question that I did.

In order to show where I’m coming from, let me present this as a combination of a draft followed by a critique.

Draft Observe that the set of polynomials with non-negative integer coefficients maps bijectively with the set of integers $\ge 2$ in a natural way, namely, that given such a polynomial, whose coefficients are a(n), … a(1), a(0), the corresponding integer is the one such that the exponent of the k-th prime in its canonical prime factorization (ie, the primes occurring in ascending sequence) is a(k-1). For example, if the coefficients are 2, 0, 1, 3, then the corresponding integer is 49 x 3 x 8 = 2976. (This bijective mapping seems to me so obvious and natural, that I can’t believe that no one has ever explored it before, and yet I don’t recall seeing any such discussion of it in the literature, but anyway, that is why I included the tag “reference-request” on this question.) Now, the roots of these polynomials may or may not be of interest, but what we want to focus on (no pun intended), is presenting the sequence consisting of the graphs of these polynomials rapidly, like the frames of in a movie film, that is, like a movie. Does this movie show any striking visual pattern? Or perhaps the n-th frame should consist not simply of the graph of the polynomials for (n + 1), but of the sum of the first n such polynomials. Anyway, does the movie show any striking visual pattern?

Critique In the draft, we ignored the issue of what the (algebraic) sign of a given coefficient should be, simply assuming that they should all be non-negative. However, bearing in mind the example of the determinant (of a square matrix), in which negative signs are alternately assigned, perhaps we should specify that, say, the coefficients of the terms of odd index be negative. Many such schemes of specifying negative coefficients are possible. Indeed, there are infinitely many possible schemes, because the scheme might not be independent of the degree of the polynomial. So, the question is, which scheme is best, in terms of generating a visual pattern when the movie is played? If we don’t know, we could pick some promising/plausible ones and then just run them and see which gives the best results.

Moreover, one may object that the visual representations need not depend on the prime factorizations, but on something else (hence the more general question that I gave in the introduction).

OK, Lights! Camera! Action!

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The map you describe is really a composition of two maps, one which sends the polynomials with non-negative coefficients to the countable direct sum of copies of $\mathbb{Z}_{\ge 0}$ (forgetting the multiplication) and the other which sends this direct sum to $\mathbb{N}$ via prime factorization. The second map is more natural than the first (the first really does not preserve many important properties of a polynomial), and the first map is really not that natural, which is probably why you haven't seen such a thing described. –  Qiaochu Yuan Jul 29 '11 at 3:15
    
There's an interesting visualization of prime factorizations at the Wolfram Demonstrations Project here: demonstrations.wolfram.com/PrimeFactorizationGraph –  tzs Jul 29 '11 at 3:21
    
@tzs: That's very pretty. –  mixedmath Jul 29 '11 at 3:40
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Arithmetic topology? arxiv.org/abs/0904.3399v1 Primes ~ knots; integers are linked knots. –  genneth Jul 29 '11 at 23:13
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up vote 2 down vote accepted

I don't really understand how your polynomials amount to a meaningful visual representation of prime factorization. The polynomial construction seems entirely forced and artificial, and in that vein doesn't appear to have any natural, salient geometric features that capture factor structure.

But your question reminds me of a mathematical note I came across, An Arithmetic Metric, which introduces a "distance between natural numbers based on their arithmetic properties, instead of their position on the real line." Essentially, if you regard multiplying (or dividing) by a prime number as a unit step, then the distance metric in the paper is given by how many steps it takes to go from one natural number to another. As an example, if $a=2^2\cdot 5$ and $b=2^5 \cdot 3$, then the distance between them is $d(a,b)=|5-2|+|1-0|+|0-1|=5$, which counts three multiplications by $2$, one multiplication by $3$, and one division by $5$ in going from $a$ to $b$.

You can then represent the natural numbers as a graph with vertices $n=1,2,3,\dots$ and all edges of the form $(n,pn)$ or $(pn,n)$ for primes $p=2,3,5,7,\dots$ and naturals $n=1,2,3,\dots$. This construction is called a Hasse diagram. If desired, one can restrict attention to only special sets of numbers sharing factors, such as $I_n = \{1,2,3,\dots,n\}$ (pictured in the Figure 1 below for $n=12$) or the divisor set $\{d: d|n\}$ (pictured in Figure 2 below for $n=60$).


Figure 1 Figure 1.


Figure 2 Figure 2.


In my opinion these types of graphs provide a satisfying and intuitive global view of the relative factor structure of naturals.

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Your answer is a good one, and since a better one hasn't come along, I'm upvoting and accepting your answer. –  Mike Jones Aug 30 '11 at 1:23
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