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
Spiral: P. Meyer's software for exploring the Ulam spiral
phenomenon in the distibution of primes.
Ulam's Spiral page, with simple viewing applet.
generalised Ulam spiral graphics
Ulam spiral page
R. Sacks' NumberSpiral page with an interesting
graphical variation on the theme
Ulam spiral page
Le village premier - Ulam spiral plotting applet
Patterns in prime numbers?
Spirals of Primes
Ulam's Prime Sprial notes
Ulam's primtalsspiral (in Danish)
Prime Spiral Applet
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
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.