What does this research mean for the field?

The maximum number of no-three-in-line points that can be selected from a modular hyperbola in the Hall-Jackson-Sudbery-Wild window is exactly 3(p-1), and its complete extremal structure depends on the number of quadratic residues. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.SUPPORTS_CONSENSUS.

What question did this study set out to answer?

To establish the exact maximum size, extremal structure, and stability of subsets of modular hyperbolas containing no three collinear points within a 2p by 2p window.

September 5, 2026Open Access

Extremal no-three-in-line subsets of a modular hyperbola in the Hall-Jackson-Sudbery-Wild window

View Full Paper

Ask AI

Bookmark

Authors

AKAleksei KudriashovNational Heritage Institute

Discussion

Member takes

Implication

Mathematical analysis proves exact upper bounds for general-position subsets of modular hyperbolas, resolving structural optimality and stability across modular grid windows.

Key Points

To establish the exact maximum size, extremal structure, and stability of subsets of modular hyperbolas containing no three collinear points within a 2p by 2p window.
Formulated linear programming certificates and an orbit lemma to determine fractional relaxations and exact point counts per window box.
Employed Bombieri's estimate along cubics and a block decomposition over runs of quadratic residues in finite fields to bound point sets in hyperbola unions.
Implemented independent automated verification pipelines across hundreds of prime instances to confirm structural theorems, line classifications, and linear programming bounds.
Proved the maximum size of a no-three-in-line subset in the modular hyperbola equals exactly 3(p−1), yielding precisely 9^s maximum configurations where s counts quadratic residues.
Established that the Hall-Jackson-Sudbery-Wild configuration is the unique maximum set if and only if p ≡ 1 (mod 4) and c is a quadratic non-residue.
Bounded the union of two modular hyperbolas to α ≤ (3.449… + o(1))(p−1) using weighted line covers and consecutive square decompositions.

Cite This Study

Aleksei Kudriashov (2026) studied this question.

synapsesocial.com/papers/6a9bd4536b95aff0620ebee1 https://doi.org/10.5281/zenodo.22275500

View Full Paper

Ask AI

Bookmark

Implication

Mathematical analysis proves exact upper bounds for general-position subsets of modular hyperbolas, resolving structural optimality and stability across modular grid windows.

Extremal no-three-in-line subsets of a modular hyperbola in the Hall-Jackson-Sudbery-Wild window

Discussion

Member takes

Implication

Key Points

Cite This Study

Extremal no-three-in-line subsets of a modular hyperbola in the Hall-Jackson-Sudbery-Wild window

Discussion

Member takes

Implication

Key Points

Cite This Study

Also Consider

Also Consider