{"id":615234,"date":"2026-06-01T12:55:33","date_gmt":"2026-06-01T12:55:33","guid":{"rendered":"https:\/\/www.newsbeep.com\/uk\/615234\/"},"modified":"2026-06-01T12:55:33","modified_gmt":"2026-06-01T12:55:33","slug":"an-openai-model-solved-a-famous-math-problem-that-stumped-humans-for-80-years","status":"publish","type":"post","link":"https:\/\/www.newsbeep.com\/uk\/615234\/","title":{"rendered":"An OpenAI model solved a famous math problem that stumped humans for 80 years"},"content":{"rendered":"<p>OpenAI\u2019s diagram is based on choosing c\u00b2 = 65, which can be satisfied by either 1\u00b2 + 8\u00b2 = 65 or 4\u00b2 + 7\u00b2 = 65. This means that if the grid spacing is 1\/\u221a65, each point will be one unit away from 16 other points: (1,8), (4,7), (7,4), (8,1), (-1,8), (-4,7), and so forth. Larger values for c\u00b2\u2014if they\u2019re chosen carefully\u2014enable more whole-number diagonals and hence more unit-distance pairs.<\/p>\n<p>However, if c\u00b2 is too large compared to the number of points in the grid, then many of the potential one-unit-away neighbors will be outside the grid.<\/p>\n<p>In short, we want to choose a c\u00b2 that\u2019s large enough but not too large. Using insights from number theory, including\u00a0<a href=\"https:\/\/en.wikipedia.org\/wiki\/Sum_of_two_squares_theorem#Jacobi&#039;s_two-square_theorem\" rel=\"nofollow noopener\" target=\"_blank\">Jacobi\u2019s two-square theorem<\/a>, Erd\u0151s was able to show that an optimally sized circle will enable the number of unit-distance pairs to grow faster than the number of points, but only barely.<\/p>\n<p>The question became \u201ccan you do better?\u201d To find an upper bound, Erd\u0151s used an argument from a quite different area of mathematics called graph theory to show that you could only have so many unit distances. But his upper bound grows much, much faster than the best lower bound he was able to construct.<\/p>\n<p>Erd\u0151s\u2019s conjecture was that the actual optimum was much closer to the lower bound than the upper one. He predicted, but couldn\u2019t prove, that the maximum number of unit-distance pairs grows just barely faster than the number of points.<\/p>\n<p>To be more precise, Erd\u0151s conjectured that the number of unit distances would be n^(1+o(1)). In other words, for a sufficiently large n, the maximum number of unit distances would be less than n^(1+\ud835\udf16) for any \ud835\udf16 &gt; 0. That could end up growing a little faster than his lower-bound construction\u2014which was n^(1 + C\/(log log n)) for some constant C\u2014but within the same general ballpark.<\/p>\n","protected":false},"excerpt":{"rendered":"OpenAI\u2019s diagram is based on choosing c\u00b2 = 65, which can be satisfied by either 1\u00b2 + 8\u00b2&hellip;\n","protected":false},"author":2,"featured_media":615235,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[20],"tags":[554,733,4308,86,56,54,55],"class_list":["post-615234","post","type-post","status-publish","format-standard","has-post-thumbnail","category-artificial-intelligence","tag-ai","tag-artificial-intelligence","tag-artificialintelligence","tag-technology","tag-uk","tag-united-kingdom","tag-unitedkingdom"],"_links":{"self":[{"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/posts\/615234","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/comments?post=615234"}],"version-history":[{"count":0,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/posts\/615234\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/media\/615235"}],"wp:attachment":[{"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/media?parent=615234"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/categories?post=615234"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/tags?post=615234"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}