---
title: 'Cannonball Polygons: Weighted Square-Sum Identities'
url: https://www.emergentmind.com/topics/cannonball-polygons
type: topic
---

# Cannonball Polygons: Weighted Square-Sum Identities

Searching arXiv for recent and directly relevant papers on cannonball polygons and related constructions.
Cannonball polygons are polygons associated with weighted square-sum identities of the form
\[
\mathcal Z^2=\sum_{i=1}^\infty w(i)i^2,
\]
where \(w:\mathbb N\to\mathbb Z_{\ge 0}\) is finitely supported, non-increasing, and bounded in sup norm by a prescribed integer \(s\), called the multiplicity. In this framework, the classical Cannonball Problem appears as the multiplicity-\(1\) case, while higher multiplicities admit a much richer existence and counting theory. The subject combines polygonal geometry, square pyramidal numbers, Diophantine representation theory, and analytic number theory; in particular, every positive integer \(\mathcal Z\) occurs as the largest side of a cannonball polygon of multiplicity \(8\), and for multiplicity \(s>8\) the number of distinct classes admits an asymptotic formula [2507.18057].

## 1. Definition and geometric data

A cannonball polygon is defined as a polygon with a distinguished vertex \(O\), integer side lengths, and the same geometric perpendicularity and nondegeneracy conditions used for arithmetic polygons, but with a different side-length rule. The required properties are as follows. Traversing the polygon starting and ending at \(O\), the first side has length \(1\); consecutive side lengths are in non-decreasing order; the number of sides of length \(1\) is at most \(s\); for each \(k\in\mathbb N\), the number of sides of length \(k\) is non-increasing as \(k\) increases; and the final side may have any integer length \(\mathcal Z\). The geometric conditions are: for each side, there is a line perpendicular to that side passing through \(O\) and one of the endpoints of the side, and no vertex has angle \(0\) or \(\pi\) [2507.18057].

The arithmetic encoding is a cannonball weight function \(w\), which satisfies four conditions: \(w(i)\ge 0\); there exists \(k\in\mathbb Z_{>0}\) such that \(w(i)=0\) for all \(i\ge k\); \(\|w\|_\infty\le s\) for some \(s\in\mathbb N\); and \(w\) is non-increasing on \([1,k]\). The corresponding square identity is
\[
\mathcal Z^2=\sum_{i=1}^\infty w(i)i^2.
\]
Geometrically, \(w(i)\) counts how many sides of length \(i\) occur in the polygon, while the final side of length \(\mathcal Z\) closes the polygon.

The parameter \(s\) is the multiplicity. In the weighted formulation this is exactly the bound
\[
\|w\|_\infty\le s.
\]
Thus multiplicity controls the maximal repetition of a side length in the polygonal realization.

## 2. Classical antecedents and relation to arithmetic polygons

The classical background is the Cannonball Problem, based on square pyramidal numbers
\[
S_n=\sum_{i=1}^n i^2=\frac{2n^3+3n^2+n}{6}.
\]
The original question asks for integer solutions of
\[
S_n=m^2.
\]
Watson proved that the only nontrivial solution is
\[
S_{24}=70^2.
\]
Within the cannonball-polygon formalism, this is precisely the multiplicity-\(1\) case: if \(\|w\|_\infty=1\), then \(w\) is a \(0\)-\(1\), non-increasing, finitely supported function, hence necessarily
\[
w(i)=1 \quad (1\le i\le n),\qquad w(i)=0 \quad (i>n),
\]
and the defining identity reduces to \(\mathcal Z^2=S_n\) [2507.18057].

A related, but distinct, precursor is the theory of arithmetic polygons. There an arithmetic polygon has integer side lengths organized around a special vertex \(O\) so that, as one traverses the polygon starting and finishing at \(O\), each side has length one greater than the preceding side; for each side there is a line perpendicular to the side passing through both \(O\) and one endpoint of that side; and there are no degenerate vertices. Arithmetic polygons are equivalent to equal sums of consecutive squares,
\[
(a+1)^2+\cdots+b^2=(b+1)^2+\cdots+c^2,
\]
which in turn are equivalent to triples of square pyramidal numbers in arithmetic progression:
\[
P_a+P_c=2P_b,\qquad P_n=\sum_{k=1}^n k^2=\frac{n(n+1)(2n+1)}{6}.
\]
That paper also discussed a possible notion of “cannonball polygon” obtained by replacing the arithmetic progression condition with side lengths \(1,2,3,\dots\), except that the final side may have any integer length. It emphasized that this analogy is only partial: there are such polygons that do not arise from any cannonball-number identity, for example one with side lengths \(1,2,\dots,36\) and final side \(52\), even though
\[
52^2=2704\neq 1^2+\cdots+36^2=16206.
\]
This establishes that the later multiplicity-based definition is not merely a rephrasing of the earlier informal analogy [2411.08398].

## 3. Weight functions, polygon classes, and realizability

The basic equivalence relation is defined by the weight function. Two cannonball polygons are in the same class if they are constructed from the same weight function \(w\). The paper states that there is a one-to-one correspondence between a class of cannonball polygons and a cannonball weight function. It also notes that each class can contain at most \(2^n\) polygons, where \(n\) is the number of sides; the different polygons in a class arise from directional choices of sides rather than from different weight data [2507.18057].

A central conversion mechanism starts from a representation by square pyramidal numbers. If
\[
m=\sum_{j=1}^r h(j)S_{n_j}\qquad\text{with}\qquad n_1<\cdots<n_r,
\]
then one defines a non-increasing weight function \(w\) by layering the summands:
\[
\begin{aligned}
w(1)=\cdots=w(n_1)&=\sum_{i=1}^r h(i),\\
w(n_1+1)=\cdots=w(n_2)&=\sum_{i=2}^r h(i),\\
&\vdots\\
w(n_{r-1}+1)=\cdots=w(n_r)&=h(r).
\end{aligned}
\]
This gives
\[
m=\sum_{i=1}^\infty w(i)i^2.
\]
Because
\[
S_n=\sum_{i=1}^n i^2,
\]
a sum of square pyramidal numbers produces a non-increasing multiplicity profile for side lengths, and that profile is exactly the data needed for a cannonball polygon.

The geometric realizability of such data is inherited from the polygonal construction. The paper states that, via the chainsaw construction from the earlier arithmetic-polygon work, every admissible weight function produces at least one polygon. A plausible implication is that the classification problem naturally splits into two layers: the arithmetic classification of admissible \(w\), and the geometric multiplicity of realizations within a fixed class.

## 4. Universal existence at multiplicity \(8\)

The first main theorem is a uniform existence result:
\[
\textbf{For any positive integer } \mathcal Z,\textbf{ there exists a cannonball polygon of multiplicity }8\textbf{ with final side of length }\mathcal Z.
\]
The argument reduces this geometric statement to an additive theorem for square pyramidal numbers:
\[
\textbf{Any positive integer }m\textbf{ can be written as a sum of at most eight square pyramidal numbers.}
\]
Setting \(m=\mathcal Z^2\) then yields a representation
\[
\mathcal Z^2=S_{n_1}+\cdots+S_{n_t}\qquad (t\le 8),
\]
from which a weight function \(w\) with \(\|w\|_\infty\le 8\) is constructed, and hence a cannonball polygon class and a polygon [2507.18057].

The proof of the eight-square-pyramidal theorem is indirect and uses a six-variable identity. Writing
\[
f(x)=\frac{2x^3+3x^2+x}{6}=S_x,
\]
the paper considers
\[
\begin{aligned}
\mathcal{S}_f(6)
&= f(x+a+1)+f(x-a)+f(x+b+1)+f(x-b)+f(x+c+1)+f(x-c)\\
&= (2x^3+6x^2+7x+3)+(4x+4)\left(\frac{a^2+a}{2}+\frac{b^2+b}{2}+\frac{c^2+c}{2}\right).
\end{aligned}
\]
This reduces the representation problem to finding \(\ell,r,x\) satisfying a size condition,
\[
0<L-f(\ell)-f(r)<6x^3,
\]
together with the congruence condition
\[
(4x+4)\mid (L-f(\ell)-f(r)).
\]
Choosing \(x=p-1\) for a suitable prime \(p\), the argument passes to a cubic congruence curve over \(\mathbb F_p\), then uses a transformation to an elliptic-curve or conic setting, the Hasse–Weil bound, Bombieri’s bound for exponential sums on curves, and an explicit result of Cobeli–Zaharescu on Lehmer points. An intermediate quantitative statement is that if
\[
m\ge 1.907\times 10^{28},
\]
then \(m\) can be written as a sum of at most eight square pyramidal numbers; the remaining finite range is handled by explicit computation [2507.18057].

The threshold \(8\) is structurally important. The paper proves universal existence for multiplicity \(8\), but it does not classify multiplicities below \(8\). It states only that multiplicity \(1\) is the classical case and that the methods establish universality at \(8\), not below it.

## 5. Asymptotic enumeration for multiplicity \(s>8\)

Let \(\mathfrak C_s(\mathcal Z)\) denote the number of distinct classes of cannonball polygons of multiplicity \(s\) and largest side \(\mathcal Z\). Because classes are identified with weight functions, \(\mathfrak C_s(\mathcal Z)\) equals the number of representations of \(\mathcal Z^2\) as a sum of \(s\) square pyramidal numbers:
\[
\mathfrak C_s(\mathcal Z)=\mathcal C_s(\mathcal Z^2).
\]
For \(s\ge 9\), the paper proves an asymptotic formula of Hardy–Littlewood type,
\[
\mathfrak C_s(\mathcal Z)
\sim
3^{s/3}\mathfrak S(\mathcal Z^2)\Gamma\!\left(\frac43\right)^s
\Gamma\!\left(\frac s3\right)^{-1}\mathcal Z^{2s/3-2},
\]
with explicit power-saving error terms [2507.18057].

The arithmetic factor is a singular series. In the general notation of the paper,
\[
\mathfrak S(m)=\sum_{q=1}^\infty V(q),
\]
where
\[
V(q):=\sum_{\substack{a=1\\(a,q)=1}}^q \bigg(\frac{V(q,a)}{6q}\bigg)^s e\!\left(-\frac{am}{q}\right),
\]
and
\[
V(q,a):=\sum_{1\leqslant n\leqslant 6q} e\bigg (\frac{a}{q}S_n \bigg).
\]
The series is multiplicative and has an Euler product
\[
\mathfrak S(m)=\prod_{p\text{ prime}} T_m(p),
\]
with
\[
T_m(p):= \lim_{k\to\infty} p^{k(1-s)}\mathcal M_m(p^k),
\]
where \(\mathcal M_m(p^k)\) counts solutions to the relevant congruence modulo \(p^k\). The positivity statement
\[
\mathfrak S(m)>0
\]
is essential, because it shows that the main term does not vanish through local congruence obstructions.

The paper gives a special explicit statement for \(s=9\). For any integer
\[
\mathcal Z\ge \exp\!\left(\frac12 e^{94}\right),
\]
it states
\[
\left|\mathfrak{C}_9(\mathcal{Z}) -\frac{4}{125}\bigg(\Gamma\bigg(\frac{4}{3}\bigg)\bigg)^{9}\mathfrak{S}_{9,S}(\mathcal{Z}^2) \mathcal{Z}^{4}\right| \leqslant 10^{14}\mathcal{Z}^{4-\frac{1}{54}},
\]
and
\[
\mathfrak{S}_{9,S}(\mathcal{Z}^2)\geqslant \exp{\left(- e^{92} \right)}.
\]
The analytic proof uses the circle method. With
\[
f_N(\alpha,S)=\sum_{n=1}^N e(\alpha S_n),\qquad
N=\left\lceil (3m)^{1/3}\right\rceil+1,
\]
the representation count is decomposed into major- and minor-arc integrals. The major arcs contribute the singular series and a singular integral with Gamma-factor asymptotics, while the minor arcs are controlled by Weyl-type bounds for cubic exponential sums, an \(L^8\)-bound, and Hölder-type estimates.

## 6. Scope, examples, and conceptual position

The theory sits at the intersection of figurate numbers, Diophantine equations, partition-like multiplicity profiles, and polygonal geometry. The paper explicitly frames square pyramidal numbers as 3D figurate numbers,
\[
S_n=\sum_{i\le n} i^2,
\]
and recasts the classical equation
\[
\frac{2n^3+3n^2+n}{6}=m^2
\]
as the special case of the weighted equation
\[
\mathcal Z^2=\sum_{i\ge 1} w(i)i^2.
\]
In this sense, cannonball polygons generalize a single exceptional Diophantine identity into a family parameterized by monotone multiplicity data [2507.18057].

A canonical example is the multiplicity-\(1\) polygon attached to Watson’s solution:
\[
\sum_{i=1}^{24} i^2=S_{24}=70^2.
\]
The paper states that a nontrivial cannonball polygon of multiplicity \(1\) can only be constructed with the final side of length \(70\). At the opposite end of the current theory, multiplicity \(8\) already guarantees existence for every \(\mathcal Z\), while multiplicities \(s\ge 9\) support asymptotic enumeration.

The limitations are equally explicit. The paper does not classify all cannonball polygons, all admissible weight functions, or the behavior of multiplicities \(2\le s\le 7\). It proves universal existence at multiplicity \(8\) and asymptotic counting for \(s>8\), but not a complete low-multiplicity theory. A plausible implication is that the central unresolved region lies between the isolated multiplicity-\(1\) phenomenon and the stable high-multiplicity regime controlled by additive and analytic methods.

Within the broader literature, cannonball polygons are best understood as a weighted successor to arithmetic polygons rather than as a synonym for them. Arithmetic polygons are governed by equal sums of consecutive squares and by triples of square pyramidal numbers in arithmetic progression, whereas cannonball polygons replace the consecutive-length condition by a non-increasing multiplicity profile. The resulting theory is therefore a distinct geometric realization of weighted square-pyramidal representations, with its own existence thresholds, equivalence classes, and asymptotic counting laws [2411.08398].

Source: https://www.emergentmind.com/topics/cannonball-polygons