Two Students Unravel a Widely Believed Math Conjecture | Quanta Magazine

Abstraction: Grad students disprove Apollonian circle packing local-global conjecture computationally

Key points:

• Summer Haag (grad student) and Clyde Kertzer (undergrad) at University of Colorado Boulder disproved the local-global conjecture for Apollonian circle packings in summer 2023
• The conjecture held that every integer in an allowed residue class (mod 24) must appear as a curvature in a given packing; it was widely assumed true after Bourgain and Kontorovich proved "virtually all" such integers appear (2012)
• Disproof was found computationally: Python scatter plots of 10,000 × 10,000 integer pairs revealed persistent black-dot patterns that should have faded
• The obstruction stems from quadratic reciprocity: certain curvature pairs violate it, meaning 24 and 8 can never appear in the same packing
• Proof posted to arXiv (arXiv:2307.02749) on July 6, 2023; supervised by Katherine Stange and James Rickards
• Result casts doubt on related conjectures including Zaremba's conjecture and Markov triple conjectures

Connections: Katherine Stange · Elena Fuchs · Number Theory · Apollonian Circle Packings · Computational Mathematics

Source: https://www.quantamagazine.org/two-students-unravel-a-widely-believed-math-conjecture-20230810/