Cannonball Polygons: Weighted Square-Sum Identities

• Cannonball polygons are defined by weighted square-sum identities linking integer side lengths with square pyramidal numbers in a geometric framework.
• The theory extends the classical cannonball problem by allowing higher multiplicities, with weight functions governing polygon classes and existence conditions.
• Multiplicities of 8 and above ensure universal existence and asymptotic counting via analytic methods, highlighting the interplay of geometry and number theory.

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 (Dong et al., 24 Jul 2025).

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 $w:\mathbb N\to\mathbb Z_{\ge 0}$0 is at most $w:\mathbb N\to\mathbb Z_{\ge 0}$1; for each $w:\mathbb N\to\mathbb Z_{\ge 0}$2, the number of sides of length $w:\mathbb N\to\mathbb Z_{\ge 0}$3 is non-increasing as $w:\mathbb N\to\mathbb Z_{\ge 0}$4 increases; and the final side may have any integer length $w:\mathbb N\to\mathbb Z_{\ge 0}$5. The geometric conditions are: for each side, there is a line perpendicular to that side passing through $w:\mathbb N\to\mathbb Z_{\ge 0}$6 and one of the endpoints of the side, and no vertex has angle $w:\mathbb N\to\mathbb Z_{\ge 0}$7 or $w:\mathbb N\to\mathbb Z_{\ge 0}$8 (Dong et al., 24 Jul 2025).

The arithmetic encoding is a cannonball weight function $w:\mathbb N\to\mathbb Z_{\ge 0}$9, which satisfies four conditions: $s$0; there exists $s$1 such that $s$2 for all $s$3; $s$4 for some $s$5; and $s$6 is non-increasing on $s$7. The corresponding square identity is

$s$8

Geometrically, $s$9 counts how many sides of length $1$0 occur in the polygon, while the final side of length $1$1 closes the polygon.

The parameter $1$2 is the multiplicity. In the weighted formulation this is exactly the bound

$1$3

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

$1$4

The original question asks for integer solutions of

$1$5

Watson proved that the only nontrivial solution is

$1$6

Within the cannonball-polygon formalism, this is precisely the multiplicity-$1$7 case: if $1$8, then $1$9 is a $\mathcal Z$0-$\mathcal Z$1, non-increasing, finitely supported function, hence necessarily

$\mathcal Z$2

and the defining identity reduces to $\mathcal Z$3 (Dong et al., 24 Jul 2025).

A related, but distinct, precursor is the theory of arithmetic polygons. There an arithmetic polygon has integer side lengths organized around a special vertex $\mathcal Z$4 so that, as one traverses the polygon starting and finishing at $\mathcal Z$5, each side has length one greater than the preceding side; for each side there is a line perpendicular to the side passing through both $\mathcal Z$6 and one endpoint of that side; and there are no degenerate vertices. Arithmetic polygons are equivalent to equal sums of consecutive squares,

$\mathcal Z$7

which in turn are equivalent to triples of square pyramidal numbers in arithmetic progression: $\mathcal Z$8 That paper also discussed a possible notion of “cannonball polygon” obtained by replacing the arithmetic progression condition with side lengths $\mathcal Z$9, 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 $8$0 and final side $8$1, even though

$8$2

This establishes that the later multiplicity-based definition is not merely a rephrasing of the earlier informal analogy (Anderson et al., 2024).

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 $8$3. 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 $8$4 polygons, where $8$5 is the number of sides; the different polygons in a class arise from directional choices of sides rather than from different weight data (Dong et al., 24 Jul 2025).

A central conversion mechanism starts from a representation by square pyramidal numbers. If

$8$6

then one defines a non-increasing weight function $8$7 by layering the summands: $8$8 This gives

$8$9

Because

$s>8$0

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 $s>8$1, and the geometric multiplicity of realizations within a fixed class.

4. Universal existence at multiplicity $s>8$2

The first main theorem is a uniform existence result: $s>8$3 The argument reduces this geometric statement to an additive theorem for square pyramidal numbers: $s>8$4 Setting $s>8$5 then yields a representation

$s>8$6

from which a weight function $s>8$7 with $s>8$8 is constructed, and hence a cannonball polygon class and a polygon (Dong et al., 24 Jul 2025).

The proof of the eight-square-pyramidal theorem is indirect and uses a six-variable identity. Writing

$s>8$9

the paper considers

$O$0

This reduces the representation problem to finding $O$1 satisfying a size condition,

$O$2

together with the congruence condition

$O$3

Choosing $O$4 for a suitable prime $O$5, the argument passes to a cubic congruence curve over $O$6, 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

$O$7

then $O$8 can be written as a sum of at most eight square pyramidal numbers; the remaining finite range is handled by explicit computation (Dong et al., 24 Jul 2025).

The threshold $O$9 is structurally important. The paper proves universal existence for multiplicity $O$0, but it does not classify multiplicities below $O$1. It states only that multiplicity $O$2 is the classical case and that the methods establish universality at $O$3, not below it.

5. Asymptotic enumeration for multiplicity $O$4

Let $O$5 denote the number of distinct classes of cannonball polygons of multiplicity $O$6 and largest side $O$7. Because classes are identified with weight functions, $O$8 equals the number of representations of $O$9 as a sum of $1$0 square pyramidal numbers: $1$1 For $1$2, the paper proves an asymptotic formula of Hardy–Littlewood type,

$1$3

with explicit power-saving error terms (Dong et al., 24 Jul 2025).

The arithmetic factor is a singular series. In the general notation of the paper,

$1$4

where

$1$5

and

$1$6

The series is multiplicative and has an Euler product

$1$7

with

$1$8

where $1$9 counts solutions to the relevant congruence modulo $w:\mathbb N\to\mathbb Z_{\ge 0}$00. The positivity statement

$w:\mathbb N\to\mathbb Z_{\ge 0}$01

is essential, because it shows that the main term does not vanish through local congruence obstructions.

The paper gives a special explicit statement for $w:\mathbb N\to\mathbb Z_{\ge 0}$02. For any integer

$w:\mathbb N\to\mathbb Z_{\ge 0}$03

it states

$w:\mathbb N\to\mathbb Z_{\ge 0}$04

and

$w:\mathbb N\to\mathbb Z_{\ge 0}$05

The analytic proof uses the circle method. With

$w:\mathbb N\to\mathbb Z_{\ge 0}$06

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 $w:\mathbb N\to\mathbb Z_{\ge 0}$07-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,

$w:\mathbb N\to\mathbb Z_{\ge 0}$08

and recasts the classical equation

$w:\mathbb N\to\mathbb Z_{\ge 0}$09

as the special case of the weighted equation

$w:\mathbb N\to\mathbb Z_{\ge 0}$10

In this sense, cannonball polygons generalize a single exceptional Diophantine identity into a family parameterized by monotone multiplicity data (Dong et al., 24 Jul 2025).

A canonical example is the multiplicity-$w:\mathbb N\to\mathbb Z_{\ge 0}$11 polygon attached to Watson’s solution: $w:\mathbb N\to\mathbb Z_{\ge 0}$12 The paper states that a nontrivial cannonball polygon of multiplicity $w:\mathbb N\to\mathbb Z_{\ge 0}$13 can only be constructed with the final side of length $w:\mathbb N\to\mathbb Z_{\ge 0}$14. At the opposite end of the current theory, multiplicity $w:\mathbb N\to\mathbb Z_{\ge 0}$15 already guarantees existence for every $w:\mathbb N\to\mathbb Z_{\ge 0}$16, while multiplicities $w:\mathbb N\to\mathbb Z_{\ge 0}$17 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 $w:\mathbb N\to\mathbb Z_{\ge 0}$18. It proves universal existence at multiplicity $w:\mathbb N\to\mathbb Z_{\ge 0}$19 and asymptotic counting for $w:\mathbb N\to\mathbb Z_{\ge 0}$20, but not a complete low-multiplicity theory. A plausible implication is that the central unresolved region lies between the isolated multiplicity-$w:\mathbb N\to\mathbb Z_{\ge 0}$21 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 (Anderson et al., 2024).