Editing Ulam spiral (section)

==Variants==
Klauber's 1932 paper describes a triangle in which row ''n'' contains the numbers (''n''  −  1)<sup>2</sup> + 1 through ''n''<sup>2</sup>. As in the Ulam spiral, quadratic polynomials generate numbers that lie in straight lines. Vertical lines correspond to numbers of the form ''k''<sup>2</sup> − ''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 [[square number|perfect square]] occurs in each full rotation. (In the Ulam spiral, two squares occur in each rotation.) Euler's prime-generating polynomial, ''x''<sup>2</sup> − ''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.)
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Additional structure may be seen when [[composite number]]s are also included in the Ulam spiral. The number 1 has only a single factor, itself; each prime number has two factors, itself and 1; 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.

Spirals following other tilings of the plane also generate lines rich in prime numbers, for example hexagonal spirals.
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<gallery>
File:KlauberTriangle.png|Klauber triangle with prime numbers generated by Euler's polynomial ''x''<sup>2</sup>  −  ''x''  +  41 highlighted.
Image:Sacks spiral.svg|Sacks spiral.
File:Spirale Ulam 150.jpg|Ulam spiral of size 150×150 showing both prime and composite numbers.
File:Hexgrid prime number spiral.svg|Hexagonal number spiral with prime numbers in green and more highly composite numbers in darker shades of blue.
File:Ulamtriangle.png|Number spiral with 7503 primes visible on regular triangle.
File:Ulam Spiral of 10 Million Primes.png|Ulam spiral with 10 million primes.
</gallery>