The Exact Value f(8) = 17 in Erdős Problem #506

We determine the exact value of the eight-point case of Erdős Problem #506, under Elliott's non-degeneracy convention that the points are neither all collinear nor all concyclic. We exhibit an explicit configuration of eight points determining exactly 17 distinct proper circles and prove that no admissible configuration can determine 16 or fewer circles. Hence f(8)=17. The accompanying verification package contains exact symbolic verification of the construction and independent computational checks of the most delicate combinatorial lemmas. This is priority version v0.1 and has not yet been peer reviewed.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-08-18
DOI
https://doi.org/10.5281/zenodo.21990624
Primary Topic
Polynomial and algebraic computation
Type
preprint
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preprint

The Exact Value f(8) = 17 in Erdős Problem #506

Mateusz Zarzycki
Zenodo (CERN European Organization for Nuclear Research)
Polynomial and algebraic computation
preprint

The Exact Value f(8) = 17 in Erdős Problem #506

Mateusz Zarzycki
preprint en

Abstract

We determine the exact value of the eight-point case of Erdős Problem #506, under Elliott's non-degeneracy convention that the points are neither all collinear nor all concyclic. We exhibit an explicit configuration of eight points determining exactly 17 distinct proper circles and prove that no admissible configuration can determine 16 or fewer circles. Hence f(8)=17. The accompanying verification package contains exact symbolic verification of the construction and independent computational checks of the most delicate combinatorial lemmas. This is priority version v0.1 and has not yet been peer reviewed.

Zenodo (CERN European Organization for Nuclear Research)
Polynomial and algebraic computation
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The Exact Value f(8) = 17 in Erdős Problem #506 — Mateusz Zarzycki · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS