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.<\/p>\n

For example, Fuchs studies Markov triples, sets of numbers that satisfy the equation x<\/em>^{2<\/sup> + y<\/em>2<\/sup> + z<\/em>2<\/sup> = 3xyz<\/em>. She and others have shown that certain types of solutions are connected for prime numbers greater than 10392<\/sup>. 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. \u201cMaybe I\u2019m missing something,\u201d she said. \u201cMaybe everyone\u2019s missing something.\u201d<\/p>\n}

\u201cNow that we have a single example where it\u2019s false, the question is: Is it false for these other examples too?\u201d Rickards said.<\/p>\n

There\u2019s also Zaremba\u2019s 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\u2019s conjecture holds for almost all numbers. But the surprise about circle packing has undermined confidence in Zaremba\u2019s conjecture.<\/p>\n

If the packing problem is a harbinger of things to come, computational data may be the tool of its undoing.<\/p>\n

\u201cI always find it fascinating when new mathematics is born out of just purely looking at data,\u201d Fuchs said. \u201cWithout it, it\u2019s really hard to imagine that [they] would have stumbled upon this.\u201d<\/p>\n

Stange added that none of this would have happened without the low-stakes summer project. \u201cSerendipity and an attitude of playful exploration both have such a huge role in discovery,\u201d she said.<\/p>\n

\u201cIt was pure coincidence,\u201d Haag said.\u00a0\u201cIf I didn\u2019t go big enough, we wouldn\u2019t have noticed it.\u201d The work bodes well for the future of number theory. \u201cYou can glean understanding of mathematics through your intuition, through proofs,\u201d Stange said. \u201cAnd you trust that a lot because you spent a lot of time thinking about it. But you can\u2019t argue with the data.\u201d<\/p>\n

Editor\u2019s note<\/em>: Alex Kontorovich is a member of <\/em>Quanta Magazine\u2019s scientific advisory board. He was interviewed for this story but did not otherwise contribute to its production.<\/em><\/p>\n<\/div>\n

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