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: In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... StanisÅ‚aw Ulam in the 1950s. ... Year 1963 (MCMLXIII) was a common year starting on Tuesday (link will display full calendar) of the Gregorian calendar. ... A doodle Look up doodle in Wiktionary, the free dictionary. ... In graph theory, a grid graph is a graph corresponding to the square lattice, so that it is isomorphic to the graph having a vertex corresponding to every pair of integers (a, b), and an edge connecting (a, b) to (a+1, b) and (a, b+1). ... This article does not cite any references or sources. ...

He then circled all of the prime numbers and he got the following picture: Image File history File links Download high resolution version (1800x1200, 103 KB) de:Zahlen von 1-50 in Spiralform angeordnet, Ausgangspunkt für die Ulam-Spirale en:Numbers from 1-50 in a spiral fr:Nombres de 1-50 en spirale sl:Števila od 1 do 50 v Ulamovem prtu...

To his surprise, the circled numbers tended to line up along diagonal lines. The image below is a 200×200 Ulam spiral, where primes are black. Diagonal lines are clearly visible, confirming the pattern. Image File history File links Download high resolution version (1800x1200, 75 KB) de:Kleine Ulam-Spirale (von 1 bis 50) en:Little Ulam spiral (from 1 to 50) fr:Petite Spirale dUlam (1 à 50) sl:Majhen Ulamov prt (od 1 do 50) Source: de:Eigene Zeichnung en:Picture made... A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ...

Ulam spiral of size 200×200

All prime numbers except 2 are odd numbers. Since in the Ulam spiral adjacent diagonals are alternatively odd and even numbers, it is no surprise that all prime 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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

Tests so far confirm that there are diagonal lines even when very large numbers of numbers are plotted. The pattern also seems to appear even if the number at the center is not 1 (and can, in fact, be 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 a number of primes as n counts up {1, 2, 3, ...} that is large by comparison with the proportion of primes among numbers of similar magnitude. This finding was so significant that the Ulam spiral appeared on the cover of Scientific American in March 1964. Scientific American is a popular-science magazine, published (first weekly and later monthly) since August 28, 1845, making it the oldest continuously published magazine in the United States. ...

At sufficient distance from the centre, horizontal and vertical lines are also clearly visible.

Variants

Variants of Ulam's spiral such as the Sacks spiral also produce intriguing and unexplained patterns. Robert Sacks came up with the Sacks spiral, a variant of an Ulam spiral in 1994. ...

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.

The American Mathematical Monthly is a mathematical journal published 10 times each year by the Mathematical Association of America since 1894. ... Martin Gardner (b. ... Scientific American is a popular-science magazine, published (first weekly and later monthly) since August 28, 1845, making it the oldest continuously published magazine in the United States. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

External links

• An applet with source code
• Software for exploring the Ulam spiral and prime patterns in other concentric polygon systems, (both spiral and non-spiral)
• Prime plots using a triangle instead of a square
• A prime spiral implementation in PHP
• The prime number cross, a similar pattern
• A Tcl/Tk version with primes and number of divisors