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The Polygonal number theorem states that every positive integer is the sum of at most $n$ $n$-gonal numbers.

I do think that expecting a "Proof without words" of this theorem is asking for something unreasonable, but is there any way to "visualize" or to get an intuitive feel for the theorem?

One motivation for this question is the problem asked by Bjorn Poonen here. If there was such an interpretation, then maybe it would help in some way to answer that question?

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    $\begingroup$ I can't even visualise the Lagrange four-squares theorem: that would seem like a sensible first step. $\endgroup$
    – James Cranch
    Commented May 13, 2011 at 9:13
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    $\begingroup$ Cauchy's proof shows that the theorem is essentially a consequence of the theorem for triangular numbers, which in turn is a special case of the 3-squares theorem, which is related to class numbers of quadratic number fields (or forms), which are related to certain (geometric) properties of quaternions; thus there might be a positive answer if you can visualize objects in 4D. But not without words: the transitions between the problems above are not exactly obvious, to say the least. $\endgroup$
    – Franz Lemmermeyer
    Commented May 13, 2011 at 10:49

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Let me make some comments on the polygonal number theorem along the theme that things are less thrilling for $k \gt 4$.

  1. Triangular and square numbers are pleasing to represent as dot patterns, after that $k$-gonal numbers are less attractive.

  2. Around 1796 Gauss proved that every integer is a sum of 3 triangular numbers. This is sometimes called the Eureka Theorem because Gauss was quite pleased with the result.

  3. Evidently, Cauchy showed (around 1813) that from that one can go to the general polygonal theorem that every integer $n$ can be represented as a sum of $k$ $k$-gonal numbers.

  4. Tables due to Peppin and Dickson show how to represent $n \lt 120k-240$ as a sum of at most $k$ $k$-gonal` numbers.

  5. A nice paper by Melyvn Nathanson uses results 2 & 4 to show in a few pages that

    a. for $k \ge 5$ and $n \ge 120k-240$, $n$ can be written as a sum of $k-1$ $k$-gonal numbers of which at most four are different than $0$ or $1$.

    b. For fixed $k$ every large enough odd $n$ is a sum of four $k$-gonal numbers and every large enough even $n$ is a sum of five $k$-gonal numbers one of which is either $0$ or $1$.

So it seems possible to me that taking 2 & 4 as given, it might be possible to express at least some of the steps of a transition to the general polygonal theorem in a somewhat geometric way, however if this hypothetical thing is possible it would probably not be a simplification giving insight as much as a somewhat more complicated path which is interesting mainly for the fact that it can be done at all.

I would think of an insight giving proof as one which also showed how to represent $n$. The proofs I know of show that there must be a representation but don't tell you how to find it. Changing $n$ to $n+1$ and/or $k$ to $k+1$ can radically change the representation.

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