Finding Line Segments in the Ulam Square with the Hough Transform

摘要

The regularities present in the Ulam spiral provided an incentive for interesting observations in the number theory. Therefore, we have made the Ulam square an object of analysis from the image processing perspective. A version of the Hough transform designed specially for detecting sequences of pixels forming segments of straight lines with the slope defined by an irreducible fraction was used to find line segments in the Ulam spiral. Angles which described the slopes of the segments had tangents p/q expressed by integers p from 0 to 10 and q from -10 to 10 (0 excluded). Due to storage limitations the squares with the side of length up to 5001 points which correspond to the largest prime 25009 991 were analyzed at present. In such a square the longest segment has 16 primes and its tangent is 3 (3 up and 1 to the right). Segments of length 14 and 15 were absent. The number of shorter segments varied strongly, from one for a 13-point segment to tens of thousands for shorter ones.