Arithmetic Polygons and Sums of Consecutive Squares

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.

💡 Research Summary

The paper introduces the notion of an “arithmetic polygon,” a polygon whose side lengths are consecutive integers and which possesses a distinguished vertex O such that each side is orthogonal to a line through O and one of its endpoints. The definition excludes degenerate angles. The classic 3‑4‑5 right triangle is the simplest example.
If the side lengths are (a+1, a+2, …, c) with an intermediate index (b) satisfying (a+1<b<c), the equality
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