A mathematical problem had been resisting experts for more than 80 years. An AI has surpassed them all
In 1946 the Hungarian mathematician Paul Erdős asked a seemingly very simple question: if you place n points in the plane, how many pairs of points can be exactly at a distance 1 from each other? This dilemma is known as unit distance problem in the planeand has maintained many mathematicians who research in the field of geometry, immersed in its resolution for no less than eighty years. The classic strategy proposed by many of them to try to solve it was to resort to a square grid. They soon realized that the number of pairs at unit distance grows at least as n to the power of (1 + C/loglog(n)), where C is a positive constant that quantifies how much a particular construction can be better than a basic square grid. It’s a complicated idea, it’s true, but we can try to approach it in a slightly more intuitive way. A standard square grid produces approximately 2n pairs of points at unit distance. If we rescale it in an ingenious way by choosing the scale factor as a number that has many divisors (in number theory this property is known as a number with many small prime factors), you get more pairs of points to fall exactly at distance 1. The value of C measures precisely the efficiency of that choice. This is the key. An AI from OpenAI has achieved the first major breakthrough in 80 years As we are seeing, the question Erdős asked is very easy to state, but extraordinarily difficult to resolve. If we develop the classical approach a little further we will realize that since loglog(n) grows very slowly, the exponent approaches 0. This means that the square grid grows only slightly faster than n, but not enough to exceed n at a fixed rate. This milestone was achieved by a general-purpose inference model that OpenAI was testing internally. This is why for decades mathematicians predicted that the upper bound would be approximately n^(1+o(1)), that is, just slightly larger than n. Now we know that they were wrong, and the person who refuted this conjecture was not a particularly skilled current mathematician; this milestone has pointed it out a general purpose inference model which OpenAI was testing internally. and not one artificial intelligence (AI) specialized in mathematics. This model has provided an infinite family of examples that produce polynomial improvement. In fact, he has shown that it is possible to construct configurations of points with at least n^(1+δ) pairs at unit distance, where δ is a fixed value greater than 0 that does not disappear as n grows. When the AI delivered this result, OpenAI researchers asked a group of Princeton mathematicians to review it. And his conclusion was blunt. The AI was right. This is the first progress on the lower bound of the problem posed by Erdős in 80 years. And, curiously, the OpenAI model has achieved this by using advanced engineering tools. algebraic number theory for an apparently elementary geometry problem. Several renowned mathematicians, such as Fields Medal winner Tim Gowers or number theory expert Arul Shankar, have declared that the result that AI has delivered is an extraordinary achievement that could provide mathematicians with a bridge to explore other problems in the future. Image | Jeswin Thomas More information | OpenAI In Xataka | These two problems have baffled mathematicians for decades. A genius has solved them with a stroke of the pen