The OpenAI model solved a well-known mathematical problem that had frustrated people for 80 years

The OpenAI image is based on the choice c² = 65, which can be satisfied by 1² + 8² = 65 or 4² + 7² = 65. This means that if the grid spacing is 1/√65, each point will be one of the 16 other points: (1,8), (4,7), (7,4), (1,8,7), (1,8,7), (1,8,7), (8,8), and Large values of c² – if chosen carefully – support integer diagonals and therefore long pairs.

However, if c² is too large compared to the number of points in the grid, then most of the neighbors with one point will be outside the grid.

In short, we want to choose c² that is large enough but not too large. Using information from number theory, including Jacobi’s two-dimensional theoremErdős was able to show that the optimally sized circle will help the number of pairs of unit distance grow faster than the number of points, but with difficulty.

The question became “can you do it right?” To find the upper limit, Erdős used an argument from a completely different area of ​​mathematics called graph theory to show that you can only have so many units. But his upper part grows faster than the lower one he built.

Erdős’ theory was that the true optimum was closer to the lower limit than the upper limit. He predicted, but could not prove, that the maximum number of pairs of distance units grows faster than the number of points.

In fact, Erdős assumed that the number of unit distances would be n^(1+o(1)). In other words, for n large enough, the maximum number of distance units would be less than n^(1+𝜖) for 𝜖 > 0. It might end up growing a little faster than its lower bound—which was n^(1 + C/(log log n)) for C at all times—but within the same ballpark.