Ulam spiral
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The Ulam spiral, or prime spiral (in other languages also called the Ulam cloth) is a simple method of graphing the prime numbers that reveals a pattern which has never been fully explained. It was discovered by the mathematician Stanisław Ulam in 1963, while doodling on scratch paper at a scientific meeting. Ulam, bored that day, wrote down a regular grid of numbers, starting with 1 at the center, and spiraling out:
He then circled all of the prime numbers and he got the following picture:
To his surprise, the circled numbers tended to line up along diagonal lines. The following image illustrates this. This is a 200×200 Ulam spiral, where primes are black. Black diagonal lines are clearly visible.
All prime numbers (except 2) are obviously odd numbers and (except for 2 and 5) end with 1, 3, 7, or 9. In the Ulam spiral adjacent diagonals are alternatively odd and even numbers, in both ways. Therefore it is no surprise that all prime numbers, being odd numbers, lie in alternate diagonals of the Ulam spiral. What is startling is the tendency of prime numbers to lie on some diagonals more than others, while a random distribution is expected.
It appears that there are diagonal lines no matter how many numbers are plotted. This seems to remain true, even if the starting number at the center is much larger than 1. This implies that there are many integer constants b and c such that the function:
- f(n) = 4n2 + bn + c
generates an unexpectedly-large number of primes as n counts up {1, 2, 3, ...}. This was so significant, that the Ulam spiral appeared on the cover of Scientific American in March 1964.
At sufficient distance from the centre, horizontal and vertical lines are also clearly visible.
[edit] Variants
Variants of Ulam's spiral also produce intriguing and unexplained patterns. One such variant was studied in 1994 by software engineer Robert Sacks. Sacks' number spiral differs from Ulam's in three ways: it places points on an Archimedian spiral rather than the square spiral used by Ulam, it places zero in the center of the spiral, and it makes a full rotation for each perfect square while the Ulam spiral places two squares per rotation. Certain curves originating from the origin appear to be unusually dense in prime numbers; one such curve, for instance, contains the numbers of the form n2 + n + 41, a famous prime-generating polynomial discovered by Leonhard Euler in 1774. The extent to which the number spiral's curves are predictive of large primes and composites remains unknown.
[edit] References
- Stein, M. and Ulam, S. M. (1967), "An Observation on the Distribution of Primes." American Mathematical Monthly 74, 43-44.
- Stein, M. L.; Ulam, S. M.; and Wells, M. B. (1964), "A Visual Display of Some Properties of the Distribution of Primes." American Mathematical Monthly 71, 516-520.
- Gardner, M. (1964), "Mathematical Recreations: The Remarkable Lore of the Prime Number." Scientific American 210, 120-128, March 1964.
- Eric W. Weisstein, Prime Spiral at MathWorld.
