Contiguous Line Segments in the Ulam Spiral: Experiments with Larger Numbers

In our previous papers we have investigated the directional structure and the numbers of straight line segments in the Ulam spiral. Our tests were limited to primes up to \(25\,009\,991\) due to memory limits. Now we have results for primes up to about \(10^9\) for the previously used directional resolution, and for the previous maximum number but with greatly increased directional resolution. For the extended resolution, new long segments have been found, among them the first one with 14 points. For larger numbers and the previous resolution, the new segments having up to 13 points were found, but the longest one is still the one with 16 points. It was confirmed that the relation of the number of segments of various lengths to the corresponding number of primes for a given integer, for large numbers, is close to linear in the double logarithmic scale.