Spiral of Ulam

Spiral of the prime numbers

In Mathematical, the spiral of Ulam , or spiral of the prime numbers (in other languages, also called clock of Ulam ) is a simple method for the representation of the prime numbers which reveals a reason which fully explained forever. She was discovered by the Mathématicien Stanislaw Marcin Ulam, at the time of a scientific conference in 1963. Ulam was wedged, constrained to listen to “ a very long talk and very tedious ”. It spent its time to crayonner and put at gribouiller consecutive entireties, starting with 1 in the center, in a species of spiral turning in the opposite direction of needles of a watch. It obtains a regular grid of numbers, starting by one 1 in the center, and spiralant towards outside like this:

Then, it surrounded all the prime numbers, it obtained the following image then:

With its surprise, the surrounded numbers tended to be aligned along diagonal lines. The following image illustrates this. It is a spiral of Ulam of 200 × 200, where the prime numbers are black. The black diagonals are clearly visible.

It appears diagonal lines comprising a quantity of traced numbers. This seems to remain true, even if the central number of the departure is larger than 1. This implies that there exist much whole constants has , B and C such as the function:

$f (N) = an^2 + bn + C \,$

generate an extraordinarily large number of prime numbers such as N contains {1, 2,3,…}. It was so significant which the spiral of Ulam appeared on the cover of Scientific American in March 1964.

With a sufficient distance from the center, the horizontal lines and verticals are also clearly visible.

For the traqueurs of prime numbers, these numbers were familiar. At the 18th century, Euler had advanced the formula $n^2 + N + 17$ which, for successive values of $n$ , gave prime numbers of $n=$ 0 to $n=$ 15. In fact, these sixteen prime numbers are those which appear on the principal diagonal of the diagram of Ulam: 17,19,23,29,37,47,59,73,89,107,127,149,173,199,227 and 257. Euler proposed another formula, $n^2 - N + 41$ , which, for successive values of $n$ between 0 and 40, produces only prime numbers. By computing, one showed that the formula of Euler $n^2 - N + 41$ was surprisingly good, since it generates prime numbers lower than ten million in 47,5% of the cases. Ulam found other formulas of which the percentages of success were almost as good as for that of Euler. With the great disappointment of in love with prime numbers like Paul Erdős, Ulam, conscious that its gribouillages did not lead to large step thing, gave up this start of imagination and turned over to give conferences on the relation between science and morality.

Spiral of the number of dividers

Another way of highlighting oblique alignments is to trace over each number, a disc of diameter equal to its number of dividers. The prime numbers thus have a disc of diameter 2. On the example opposite, the prime numbers are in reds and the blue discs represent the number of dividers of the numbers of which they are the center.

External bonds

http://villemin.gerard.free.fr/Wwwgvmm/Premier/Ulam.htm Introduction to the prime numbers, the spiral of Ulam.
http://www.phpcs.com/codes/SPIRALE-ULAM-NOMBRES-PREMIERS_42487.aspx to easily draw a spiral of Ulam
http://www.maths.ex.ac.uk/~mwatkins/zeta/ulam.htm Bonds towards the pages concerning the spiral of Ulam (in English).
http://yoyo.cc.monash.edu.au/~bunyip/primes/primeSpiral.htm Beautiful images (in English).
http://www.alpertron.com.ar/ULAM.HTM an applet which draws the spirals of various sizes (in English).
http://cdl.best.vwh.net/Java/PrimeSpiralApplet.html an applet with its source code (in English).
http://teachers.crescentschool.org/weissworld/m3/spiral/spiral.html wide Theory ramifying starting from the spiral, implying the prime numbers (in English).
http://www.abarim-publications.com/artctulam.html Beautiful image on the first page, theory extended on the second (in English).
http://wfr.tcl.tk/305 a version in Tcl/Tk with a generalization according to the number of dividers.

References

Stein, Mr. and Ulam, S. Mr. (1967), “Year Observation one the Distribution off Premiums. ” Bitter. Maths. Monthly 74,43-44.
Stein, Mr. L.; Ulam, S. Mr.; and Wells, Mr. B. (1964), “has Visual Display off Properties Nap off the Distribution off Premiums. ” Bitter. Maths. Monthly 71,516-520.
Gardner, Mr. (1964), “Mathematical Re-creations: The Remarkable Lore off the Number Premium. ” Sci. Land-mark. 210,120-128, March 1964.
Paul Hoffman: Erdős, the man who liked only the numbers (Editions BELIN, 2000 - ISBN 2-7011-2539-1)

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