They looked exactly as expected: a wall of white, peppered with black specks for smaller integers. “We expected the black dots to peter out,” Stange said. Rickards added, “I thought maybe it would even be possible to prove they peter out.” He speculated that by looking at charts that synthesized many packings together, the team would be able to prove results that weren’t possible when they looked at any one packing on its own.
While Stange was away, Haag wound up plotting every pair of remainders — about 120. No surprises there. Then she went big.
Haag had been plotting how 1,000 integers interact. (The graph is bigger than it sounds, since it involves 1 million possible pairs.) Then she cranked the dial up to 10,000 times 10,000. In one graph, regular rows and columns of black specks refused to dissolve. It looked nothing like what the local-global conjecture would predict.
The team met on a Monday after Stange returned. Haag presented her graphs, and they all focused on the one with the weird dots. “It was just a continual pattern,” Haag said. “And that was when Kate said, ‘What if the local-global conjecture isn’t true?’”
“This looks like a pattern. It has to continue. So the local-global conjecture must be false,” Stange recalled thinking. “James was more skeptical.”
“My first thought was there must be a bug in my code,” Rickards said. “I mean, that was the only reasonable thing I could think of.”
Within half a day, Rickards came around. The pattern ruled out all pairs where the first number is of the form 8 × (3n ± 1)2 and the second is 24 times any square. This means 24 and 8 never appear in the same packing. Numbers you’d expect to occur don’t.
“I was kind of giddy. It’s not very often that something really surprises you,” Stange said. “But that’s the magic of playing with data.”
The July paper outlines a rigorous proof that the pattern they observed continues indefinitely, disproving the conjecture. The proof hinges on a centuries-old principle called quadratic reciprocity that involves the squares of two prime numbers. Stange’s team discovered how reciprocity applies to circle packings. It explains why certain curvatures can’t be tangent to each other. The rule, called an obstruction, propagates throughout the whole packing. “It’s just an entirely new thing,” said Jeffrey Lagarias, a mathematician at the University of Michigan who was a co-author on the 2003 circle-packing paper. “They’ve found it ingeniously,” Sarnak said. “If these numbers did appear, they would violate reciprocity.”
The Fallout
A number of other conjectures in number theory may now be in doubt. Like the local-global conjecture, they are hard to prove but have already been shown to hold for virtually all cases and are generally assumed to be true.
For example, Fuchs studies Markov triples, sets of numbers that satisfy the equation x2 + y2 + z2=3xyz. She and others have shown that certain types of solutions are connected for prime numbers greater than 10392. Everyone believes the pattern should continue to infinity. But in light of the new result, Fuchs has allowed herself to feel a twinge of doubt. “Maybe I’m missing something,” she said. “Maybe everyone’s missing something.”
“Now that we have a single example where it’s false, the question is: Is it false for these other examples too?” Rickards said.
There’s also Zaremba’s conjecture. It says that a fraction with any denominator can be expressed as a continued fraction that uses only the numbers between 1 and 5. In 2014, Kontorovich and Bourgain showed that Zaremba’s conjecture holds for almost all numbers. But the surprise about circle packing has undermined confidence in Zaremba’s conjecture.
If the packing problem is a harbinger of things to come, computational data may be the tool of its undoing.
“I always find it fascinating when new mathematics is born out of just purely looking at data,” Fuchs said. “Without it, it’s really hard to imagine that [they] would have stumbled upon this.”
Stange added that none of this would have happened without the low-stakes summer project. “Serendipity and an attitude of playful exploration both have such a huge role in discovery,” she said.
“It was pure coincidence,” Haag said. “If I didn’t go big enough, we wouldn’t have noticed it.” The work bodes well for the future of number theory. “You can glean understanding of mathematics through your intuition, through proofs,” Stange said. “And you trust that a lot because you spent a lot of time thinking about it. But you can’t argue with the data.”
Editor’s note: Alex Kontorovich is a member of Quanta Magazine’s scientific advisory board. He was interviewed for this story but did not otherwise contribute to its production.
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