{"id":37189,"date":"2023-08-11T21:58:18","date_gmt":"2023-08-11T21:58:18","guid":{"rendered":"https:\/\/scienceandnerds.com\/2023\/08\/11\/two-students-unravel-a-widely-believed-math-conjecture\/"},"modified":"2023-08-11T21:58:19","modified_gmt":"2023-08-11T21:58:19","slug":"two-students-unravel-a-widely-believed-math-conjecture","status":"publish","type":"post","link":"https:\/\/scienceandnerds.com\/2023\/08\/11\/two-students-unravel-a-widely-believed-math-conjecture\/","title":{"rendered":"Two Students Unravel a Widely Believed Math Conjecture"},"content":{"rendered":"

Source:https:\/\/www.quantamagazine.org\/two-students-unravel-a-widely-believed-math-conjecture-20230810\/#comments<\/a><\/br>
\nTwo Students Unravel a Widely Believed Math Conjecture<\/br>
\n2023-08-11 21:58:18<\/br><\/p>\n

\n

They looked exactly as expected: a wall of white, peppered with black specks for smaller integers. \u201cWe expected the black dots to peter out,\u201d Stange said. Rickards added, \u201cI thought maybe it would even be possible to prove they peter out.\u201d He speculated that by looking at charts that synthesized many packings together, the team would be able to prove results that weren\u2019t possible when they looked at any one packing on its own.<\/p>\n

While Stange was away, Haag wound up plotting every pair of remainders \u2014 about 120. No surprises there. Then she went big.<\/p>\n

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.<\/p>\n

The team met on a Monday after Stange returned. Haag presented her graphs, and they all focused on the one with the weird dots. \u201cIt was just a continual pattern,\u201d Haag said. \u201cAnd that was when Kate said, \u2018What if the local-global conjecture isn\u2019t true?\u2019\u201d<\/p>\n

\u201cThis looks like a pattern. It has to continue. So the local-global conjecture must be false,\u201d Stange recalled thinking. \u201cJames was more skeptical.\u201d<\/p>\n

\u201cMy first thought was there must be a bug in my code,\u201d Rickards said. \u201cI mean, that was the only reasonable thing I could think of.\u201d<\/p>\n

Within half a day, Rickards came around. The pattern ruled out all pairs where the first number is of the form 8 \u00d7 (3n<\/em> \u00b1 1)2<\/sup> and the second is 24 times any square. This means 24 and 8 never appear in the same packing. Numbers you\u2019d expect to occur don\u2019t.<\/p>\n

\u201cI was kind of giddy. It\u2019s not very often that something really surprises you,\u201d Stange said. \u201cBut that\u2019s the magic of playing with data.\u201d<\/p>\n

The July paper<\/a> 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\u2019s team discovered how reciprocity applies to circle packings. It explains why certain curvatures can\u2019t be tangent to each other. The rule, called an obstruction, propagates throughout the whole packing. \u201cIt\u2019s just an entirely new thing,\u201d said Jeffrey Lagarias<\/a>, a mathematician at the University of Michigan who was a co-author on the 2003 circle-packing paper. \u201cThey\u2019ve found it ingeniously,\u201d Sarnak said. \u201cIf these numbers did appear, they would violate reciprocity.\u201d<\/p>\n

The Fallout<\/strong><\/h2>\n

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

<\/br><\/br><\/br><\/p>\n

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\nSource:https:\/\/www.quantamagazine.org\/two-students-unravel-a-widely-believed-math-conjecture-20230810\/#comments<\/a><\/br><\/br><\/p>\n","protected":false},"excerpt":{"rendered":"

Source:https:\/\/www.quantamagazine.org\/two-students-unravel-a-widely-believed-math-conjecture-20230810\/#comments Two Students Unravel a Widely Believed Math Conjecture 2023-08-11 21:58:18 They looked exactly as expected: a wall of white, peppered with black specks for smaller integers. \u201cWe expected the black dots to peter out,\u201d Stange said. Rickards added, \u201cI thought maybe it would even be possible to prove they peter out.\u201d He speculated that […]<\/p>\n","protected":false},"author":1,"featured_media":37190,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","om_disable_all_campaigns":false,"pagelayer_contact_templates":[],"_pagelayer_content":"","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-37189","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"yoast_head":"\nTwo Students Unravel a Widely Believed Math Conjecture - Science and Nerds<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/scienceandnerds.com\/2023\/08\/11\/two-students-unravel-a-widely-believed-math-conjecture\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Two Students Unravel a Widely Believed Math Conjecture - Science and Nerds\" \/>\n<meta property=\"og:description\" content=\"Source:https:\/\/www.quantamagazine.org\/two-students-unravel-a-widely-believed-math-conjecture-20230810\/#comments Two Students Unravel a Widely Believed Math Conjecture 2023-08-11 21:58:18 They looked exactly as expected: a wall of white, peppered with black specks for smaller integers. \u201cWe expected the black dots to peter out,\u201d Stange said. 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