Arithmetic Polygons and Sums of Consecutive Squares

Abstract

We introduce and study arithmetic polygons. We show that these arithmetic polygons are connected to triples of square pyramidal numbers. For every odd $N\geq3$, we prove that there is at least one arithmetic polygon with $N$ sides. We also show that there are infinitely many arithmetic polygons with an even number of sides.

Figures (6)

• Figure 1.1: Two examples of cannonball polygons. In both examples, the side lengths increase as one travels clockwise around the polygon starting from $O$.
• Figure 3.1: Constructing an arithmetic polygon from the $(9, 12, 14)$ solution.
• Figure 3.2: Constructing an arithmetic polygon from the $(464, 480, 495)$ solution using the chainsaw process.
• Figure 3.3: Two examples of polygons constructed from the chainsaw process.
• Figure 3.4: In both cases, we see that vertex $Q'_{c-1}$ lies within $\mathcal{S}$, giving two possibilities for $Q'_c$ which are both above the $x$-axis.

Theorems & Definitions (26)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark
• Theorem 1.5
• Theorem 2.1
• proof
• Lemma 2.2
• proof