The Ulam spiral phenomenon

There is currently no explanation for the distinct diagonal lines which appear when the primes are marked out along a particular 'square spiral' path. This was accidentally discovered by nuclear physicist Stanislaw Ulam while he was passing the time during a boring lecture.

Prime Number Spiral: P. Meyer's software for exploring the Ulam spiral phenomenon in the distibution of primes.

Dario Alpern's Ulam's Spiral page, with simple viewing applet.

J.-F. Collona's generalised Ulam spiral graphics

A. Leatherland's Ulam spiral page

R. Sacks' NumberSpiral page with an interesting graphical variation on the theme

Michel Charpentier's Ulam spiral page

Le village premier - Ulam spiral plotting applet

Wolfgang Schildbach's Patterns in prime numbers?

Bryan Clair's Spirals of Primes

Harvey Heinz's Ulam's Prime Sprial notes

Birger Nielsen's Ulam's primtalsspiral (in Danish)

C. Lane's Prime Spiral Applet

A. Uittenbogaard's thoughts on the Ulam phenomenon

R. Turco, "The secret of Ulam's spiral, the forms 6k+1 and the Goldbach's problem"

S. Nielsen, PRIMEPATTERNS (e-book, 2010)


Stein, Ulam and Wells, "A visual display of some properties of the distribution of primes", American Mathematics Monthly 71 (5) 516-520.

A.K. Dewdney, "How to pan for primes in numerical gravel", Scientific American, July 1988, p.90-93.

S.M. Ellerstein, "The Pronic Renaissance: The Ulam Square Spiral (Modified)" Journal of Recreational Mathematics 29 (3), 1998, p. 188-189.

S.M. Ellerstein, "The Pronic Renaissance II: The Ellerstein Square Spiral", Journal of Recreational Mathematics 30 (4), 1999-2000, p. 246-250.

According to E. Weisstein's Mathworld site,

"Remarkably, noted science fiction author Arthur C. Clarke described the prime spiral in his novel The City and the Stars (1956, Ch. 6, p. 54). Clarke wrote, "Jeserac sat motionless within a whirlpool of numbers. The first thousand primes.... Jeserac was no mathematician, though sometimes he liked to believe he was. All he could do was to search among the infinite array of primes for special relationships and rules which more talented men might incorporate in general laws. He could find how numbers behaved, but he could not explain why. It was his pleasure to hack his way through the arithmetical jungle, and sometimes he discovered wonders that more skillful explorers had missed. He set up the matrix of all possible integers, and started his computer stringing the primes across its surface as beads might be arranged at the intersections of a mesh."

However, Clarke never actually performed this thought experiment (pers. comm. to E. Pegg Jr., May 27, 2002), thus leaving discovery of the unexpected properties of the prime spiral to Ulam."

More about this, including an explanatory comment from Clarke himself can be found here.

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