Apollonian Circles: A Refutation of a Large-scale Hypothesis

And in 2012, Alex Kontorowicz and mathematician Jan Burgin, who died in 2018, proved that, that almost all of the numbers predicted by the guess actually occur in circular packages. But “almost all” does not mean “all”. Square numbers, for example, are so rare that, mathematically speaking, “almost” all integers are not square numbers, although we know enough of counterexamples like 25 and 49. Experts believed that the rare counter-examples that were still possible after the work of Kontorovich and Burgin would no longer exist. The main reason for this, Kontorowicz explains, is that two or three of the best-studied circular enclosures seemed to fulfill the local-global guess quite well.

Promising summer project

When Haag and Kurtzer begin their research project in Boulder in the summer of 2023, Rickards writes topic ideas on the chalkboard in Stange’s office. “We had a complete roster,” Rickards said. They had four or five starting points for the experiment. “Things you can play with and see what happens.”

One idea was to calculate all possible circular squiggles containing any two curves A and B. Rickards wrote a program that prints out what integers show when A is included as a curvature. On the basis of this program, Haag created an algorithm that graphically displays several simulations simultaneously. It was like a multiplication table: Haag chose which rows and columns to include based on the remainder when she divided the curvatures of the circles by 24. Pairs of numbers that appeared together in an Apollonian bundle received white pixels, and those that did not appeared black. Haag worked on dozens of graphs, one for each pair of residues in each of the six groups.

The results looked exactly as expected: a white wall filled with lots of black spots for smaller integers. “We expected the black dots to disappear for the larger numbers,” Stange said. By looking at graphs that summarize the characteristics of many beams, the team hoped to show results that wouldn’t be possible by looking at a single beam alone. When Stange was on a research trip, Haag charted for the first time how about 120 pairs of residual values ​​interact. There were no surprises there. But then she devoted herself to other husbands.