Ulam Spiral and Prime-Rich Polynomials

摘要

The set of prime numbers visualized as Ulam spiral was considered from the image processing perspective. Sequences of primes forming line segments were detected with the special version of the Hough transform. The polynomials which generate the numbers related to these sequences were investigated for their potential richness in prime numbers. One of the polynomials which generates the numbers forming the 11-point sequence was found exceptionally prime-rich, although it was not the longest sequence found. This polynomial is 4n~2 - 1260n+ 98827 and it generates 613 primes (20 of them with the minus sign) for the first 1000 non-negative integers as arguments. This is more than generated by some other well-known prime-rich polynomials, the Euler one included.