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)∈{6,8,18} and characterize each value by conditions on Gaussian and Eisenstein integers. For every integer N≥2, on the other hand, a(Ne)→∞ and limsupe→∞a(Ne)/e≥2√2; a quadratic chord tower combined with Farey neighbors gives explicit polygons, for instance a(8192)≥38. We also show that an odd-sided polygon of radius n exists if and only if n has a prime divisor congruent to 1pmod4, and that a(n) is even whenever at most one distinct prime divisor of n is congruent to 1pmod4; it is also even when the Gaussian part of n is the product of two distinct primes congruent to 1pmod4. Moreover, a(Ne)=ON(eω(N)) for every fixed N≥2, and a(qe)=Θ(e) for primes q≡1pmod4; 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.
Related
- Proof of the local version of the Pólya--Szegö conjecture for the torsional rigidity of polygons
- Computer-assisted local maximality of regular polygons for torsional rigidity
- Indefinite integrals of Bessel and Struve functions with half-integer indices, and incomplete gamma functions with integer indices, all multiplied by Exp(−ax2) and divided by powers
- From scalar rigidity to abstract prime set and abstract integer system
- Quadratic growth rigidity for the complex Monge--Ampère equation
BibTeX
@misc{huber2026semiprime,
title = {{Semiprime Rigidity and Growth along Powers in Integer-Sided Cyclic Polygons}},
author = {Felix Huber},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02712},
eprint = {2611.02712},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02712},
}
RIS
TY - UNPB
TI - Semiprime Rigidity and Growth along Powers in Integer-Sided Cyclic Polygons
AU - Felix Huber
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02712
UR - https://arxiv.org/abs/2611.02712
AB - 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)∈{6,8,18} and characterize each value by conditions on Gaussian and Eisenstein integers. For every integer N≥2, on the other hand, a(N^e)→∞ and limsup_e→∞a(N^e)/e≥2√2; a quadratic chord tower combined with Farey neighbors gives explicit polygons, for instance a(8192)≥38. We also show that an odd-sided polygon of radius n exists if and only if n has a prime divisor congruent to 1pmod4, and that a(n) is even whenever at most one distinct prime divisor of n is congruent to 1pmod4; it is also even when the Gaussian part of n is the product of two distinct primes congruent to 1pmod4. Moreover, a(N^e)=O_N(e^ω(N)) for every fixed N≥2, and a(q^e)=Θ(e) for primes q≡1pmod4; 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.
ER -