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Paul Erd\u0151s made many conjectures about numbers in his life<\/p>\n
Oliver Helbig\/Getty Images<\/p>\n<\/p>\n
Just a week after an AI disproved<\/a> an 80-year-old conjecture and astonished mathematicians, another conjecture that had stood for half a century has fallen, inspired by the same techniques, but this time written entirely by humans.<\/p>\n Last week, an unreleased AI model from OpenAI disproved an important conjecture first posed by Hungarian mathematician Paul Erd\u0151s<\/a>, called the unit distance problem. The puzzle, which Erd\u0151s considered his \u201cmost striking contribution to geometry\u201d and which many mathematicians had failed to unravel, concerns the number of similar-sized connections you can make between dots arranged on a flat surface.<\/p>\n Erd\u0151s had set an upper ceiling on this number, which many experts had assumed was correct. But the AI model showed that this number could in fact be much larger, using an obscure trick from algebraic number theory to make complex structures with extremely high dimensions, which could then be used to arrange the dots in a very different arrangement than humans had considered. The result took mathematicians by surprise, with some not expecting to see Erd\u0151s\u2019s conjecture disproved in their lifetimes.<\/p>\n Now, less than a week later, Thomas Bloom<\/a> at the University of Manchester in the UK and his colleagues have used a similar argument to disprove another famous claim, which Erd\u0151s had first posed in 1976, called the sum-product conjecture.<\/p>\n \u201cIt was a surprise because I had thought about the problem quite a bit,\u201d says Bloom. After seeing the trick used by OpenAI\u2019s AI, which used number theory to solve a geometric problem, Bloom and his team realised that they could try the same thing for the sum-product conjecture. \u201cOnce you know that something might be possible, you\u2019re willing to try a bit harder to actually get it to work,\u201d he says.<\/p>\n Erd\u0151s\u2019s sum-product conjecture concerns collections of numbers, or sets. It says that if you either add or multiply all the numbers together in this set, one pair at a time, to create a further two sets, then at least one of these sets must be much larger than the original set \u2013 you can\u2019t have both sets similarly small. For instance, if you multiply all the numbers from 1 through 5, you will have a larger set than if you add them all, because there will be duplicate results, such as 2+3 and 1+4. Considering a different set, such as 1, 2, 4, 8 and 16, the added set will instead be larger, because the multiplied set just contains various powers of two.<\/p>\n Erd\u0151s set a bar for how small the larger of the two added and multiplied sets could be, and conjectured this should hold for any set of numbers. But Bloom and his colleagues used the same high-dimensional trick to find a set where both its sum and multiplied are smaller than Erd\u0151s thought possible. Instead of using a geometric progression of numbers, like powers of two, you can create a progression of numbers in many different dimensions at the same time, which they found produces a set where the number of different sums you can make is much smaller.<\/p>\n \u201cThe real surprise for me was that it was so simple,\u201d says Bloom. \u201cThe construction is so simple to describe and we do genuinely understand now why [Erd\u0151s\u2019s conjecture] fails, which should help us with lots of other related problems as well.\u201d<\/p>\n \u201cThis is typical for maths as a competitive sport,\u201d says Misha Rudnev<\/a> at the University of Bristol, UK. \u201cAs soon as a new idea kicks in, some people are ready to work twenty-four hours to find more applications to it, and these people are usually very good and quick.\u201d<\/p>\n Rudnev says that Erd\u0151s\u2019s original intuition was that this conjecture should mainly be true for integers, or whole numbers, and that still appears to be true, because the set found by Bloom and his team used exotic number systems that get ever more complicated as their sets grow larger. Bloom agrees that the conjecture still holds for integers, and that \u201cthere\u2019s still a huge amount of work to be done; we don\u2019t really understand what\u2019s going on.\u201d<\/p>\n The main insight from the proof is that problems that seem geometric, such as sets of square powers of two, can actually be tackled with tools from number theory, says Bloom. \u201cIt really opens these problems to a whole new community as well. People in algebraic number theory weren\u2019t really engaging with these questions.\u201d<\/p>\n Topics:<\/p>\n\n