Semiprime Rigidity and Growth along Powers in Integer-Sided Cyclic Polygons

Abstract

For a positive integer $n$, let $a(n)$ be the largest number of sides of a simple cyclic polygon of circumradius $n$ whose side lengths are positive integers. For primes $p,q$ we show that $a(pq)\in\{6,8,18\}$ and characterize each value by conditions on Gaussian and Eisenstein integers. For every integer $N\ge2$, on the other hand, $a(N^e)\to\infty$ and $\limsup_{e\to\infty}a(N^e)/e\ge2\sqrt2$; a quadratic chord tower combined with Farey neighbors gives explicit polygons, for instance $a(8192)\ge38$. We also show that an odd-sided polygon of radius $n$ exists if and only if $n$ has a prime divisor congruent to $1\pmod4$, and that $a(n)$ is even whenever at most one distinct prime divisor of $n$ is congruent to $1\pmod4$; it is also even when the Gaussian part of $n$ is the product of two distinct primes congruent to $1\pmod4$. Moreover, $a(N^e)=O_N(e^{ω(N)})$ for every fixed $N\ge2$, and $a(q^e)=Θ(e)$ for primes $q\equiv1\pmod4$; exceptional semiprime extremisers have unique side-length multisets. The proofs use a splitting lemma for angle relations by squarefree class, arithmetic in imaginary quadratic fields, bounded zero-sum problems, and a one-dimensional Farey argument. No computer-assisted step is used.