Ulam Spiral - Variants

Variants

Klauber's 1932 paper describes a triangle in which row n contains the numbers (n-1)2+1 through n2. As in the Ulam spiral, quadratic polynomials generate numbers that lie in straight lines. Vertical lines correspond to numbers of the form k2−k+M. Vertical and diagonal lines with a high density of prime numbers are evident in the figure.

Robert Sacks devised a variant of the Ulam spiral in 1994. In the Sacks spiral the non-negative integers are plotted on an Archimedean spiral rather than the square spiral used by Ulam, and are spaced so that one perfect square occurs in each full rotation. (In the Ulam spiral, two squares occur in each rotation.) Euler's prime-generating polynomial, x2−x+41, now appears as a single curve as x takes the values 0, 1, 2, ... This curve asymptotically approaches a horizontal line in the left half of the figure. (In the Ulam spiral, Euler's polynomial forms two diagonal lines, one in the top half of the figure, corresponding to even values of x in the sequence, the other in the bottom half of the figure corresponding to odd values of x in the sequence.)

Additional structure may be seen when composite numbers are also included in the Ulam spiral. Prime numbers are divisible only by themselves and the number 1 whereas composite numbers are divisible by at least three different factors. Using the size of the dot representing an integer to indicate the number of factors and coloring prime numbers red and composite numbers blue produces the figure shown.