Ulam spiral

Ulam spiral of size 200×200. Black dots represent prime numbers. Diagonal, vertical, and horizontal lines with a high density of prime numbers are clearly visible.

The Ulam spiral or prime spiral (in other languages also called the Ulam cloth) is a simple method of visualizing the prime numbers that reveals the apparent tendency of certain quadratic polynomials to generate unusually large numbers of primes. It was discovered by the mathematician Stanislaw Ulam in 1963, while he was doodling during the presentation of a "long and very boring paper"^[1] at a scientific meeting. Shortly afterwards, in an early application of computer graphics, Ulam with collaborators Myron Stein and Mark Wells used MANIAC II at Los Alamos Scientific Laboratory to produce pictures of the spiral for numbers up to 65,000.^[2]^[1]^[3] In March of the following year, Martin Gardner wrote about the Ulam spiral in his Mathematical Games column;^[1] the Ulam spiral featured on the front cover of the issue of Scientific American in which the column appeared.

In an addendum to the Scientific American column,^[4] Gardner mentions work of the herpetologist Laurence M. Klauber on two dimensional arrays of prime numbers for finding prime-rich quadratic polynomials which was presented at a meeting of the Mathematical Association of America in 1932—more than thirty years prior to Ulam's discovery. Unlike Ulam's array, Klauber's was not a spiral. Its shape was also triangular rather than square.^[5]

Construction

Ulam constructed the spiral by writing down a regular rectangular 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. In the 200×200 Ulam spiral shown above, diagonal lines are clearly visible, confirming the pattern. Horizontal and vertical lines, while less prominent, are also evident.

All prime numbers, except for the number 2, are odd numbers. Since in the Ulam spiral either all numbers in a diagonal are odd, or all numbers are even, and adjacent diagonals alternate between odd and even, 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.

Tests so far confirm that there are diagonal lines even when many 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) = 4 n^2 + b n + c

generates, as n counts up {1, 2, 3, ...}, a number of primes that is large by comparison with the proportion of primes among numbers of similar magnitude.

In a passage from his 1956 novel The City and the Stars, author Arthur C. Clarke describes the prime spiral seven years before it was discovered by Ulam. Clarke did not notice the pattern revealed by the prime spiral because he never actually performed the experiment.^[6]

Hardy and Littlewood's Conjecture F

In their 1923 paper on the Goldbach Conjecture, Hardy and Littlewood stated a series of conjectures, one of which, if true, would explain some of the striking features of the Ulam spiral. This conjecture, which Hardy and Littlewood called "Conjecture F", is a special case of the Bateman–Horn conjecture and asserts an asymptotic formula for the number of primes of the form ax² + bx + c. Rays emanating from the central region of the Ulam spiral making angles of 45° with the horizontal and vertical correspond to numbers of the form 4x² + bx + c with b even; horizontal and vertical rays correspond to numbers of the same form with b odd. Conjecture F provides a formula that can be used to estimate the density of primes along such rays. It implies that there will be considerable variability in the density along different rays. In particular, the density is highly sensitive to the discriminant of the polynomial, b² − 16c.

The primes of the form 4x² − 2x + 41 with x = 0, 1, 2, ... have been highlighted in minority color. The prominent parallel line in the lower half of the figure corresponds to 4x² + 2x + 41 or, equivalently, to negative values of x.

Conjecture F is concerned with polynomials of the form ax² + bx + c where a, b, and c are integers and a is positive. If the coefficients contain a common factor greater than 1 or if the discriminant Δ = b² − 4ac is a perfect square, the polynomial factorizes and therefore produces composite numbers as x takes the values 0, 1, 2, ... (except possibly for one or two values of x where one of the factors equals 1). Moreover, if a + b and c are both even, the polynomial produces only even values, and is therefore composite except possibly for the value 2. Hardy and Littlewood assert that, apart from these situations, ax² + bx + c takes prime values infinitely often as x takes the values 0, 1, 2, ... This statement is a special case of an earlier conjecture of Bunyakovsky and remains open. Hardy and Littlewood further assert that, asymptotically, the number P(n) of primes of the form ax² + bx + c and less than n is given by

P(n)\sim A\frac{1}{\sqrt{a}}\frac{\sqrt{n}}{\log n}

where A depends on a, b, and c but not on n. By the prime number theorem, this formula with A set equal to one is the asymptotic number of primes less than n expected in a random set of numbers having the same density as the set of numbers of the form ax² + bx + c. But since A can take values bigger or smaller than 1, some polynomials, according to the conjecture, will be especially rich in primes, and others especially poor. An unusually rich polynomial is 4x² − 2x + 41 which forms a visible line in the Ulam spiral. The constant A for this polynomial is approximately 6.6, meaning that the numbers it generates are almost seven times as likely to be prime as random numbers of comparable size, according to the conjecture. This particular polynomial is related to Euler's prime-generating polynomial x² − x + 41 by replacing x with 2x, or equivalently, by restricting x to the even numbers. Hardy and Littlewood's formula for the constant A is

A=\varepsilon \prod _{p}{\biggl (}{\frac {p}{p-1}}{\biggr )}\,\prod _{\varpi }{\biggl (}1-{\frac {1}{\varpi -1}}{\Bigl (}{\frac {\Delta }{\varpi }}{\Bigr )}{\biggr )}.

A simpler, but obviously equivalent formula is given by:

$A=\prod \limits _{{p}}{\frac {1-{\frac {\omega (p)}{p}}}{1-{\frac {1}{p}}}}~$ , where $p$ runs over all primes, and $\omega (p)$ — is number of zeros of the quadratic polynomials modulus 'p'.

It further simplifies to $\prod \limits _{p}{\frac {p-\omega (p)}{p-1}}~$ .

In the first formula, explanation is a little bit more complex. There, in the first product, p is an odd prime dividing both a and b; in the second product, $\varpi$ is an odd prime not dividing a. The quantity ε is defined to be 1 if a + b is odd and 2 if a + b is even. The symbol $\left(\frac{\Delta}{\varpi}\right)$ is the Legendre symbol. A quadratic polynomial with A ≈ 11.3, currently the highest known value, has been discovered by Jacobson and Williams.^[7]^[8]

Variants

Klauber's 1932 paper describes a triangle in which row n contains the numbers (n − 1)² + 1 through n². As in the Ulam spiral, quadratic polynomials generate numbers that lie in straight lines. Vertical lines correspond to numbers of the form k² − k + M. Vertical and diagonal lines with a high density of prime numbers are evident in the figure.

Robert Sacks devised a variant of the Ulam spiral in 1994. In the Sacks spiral, the non-negative integers are plotted on an Archimedean spiral rather than the square spiral used by Ulam, and are spaced so that one perfect square occurs in each full rotation. (In the Ulam spiral, two squares occur in each rotation.) Euler's prime-generating polynomial, x² − x + 41, now appears as a single curve as x takes the values 0, 1, 2, ... This curve asymptotically approaches a horizontal line in the left half of the figure. (In the Ulam spiral, Euler's polynomial forms two diagonal lines, one in the top half of the figure, corresponding to even values of x in the sequence, the other in the bottom half of the figure corresponding to odd values of x in the sequence.)

Additional structure may be seen when composite numbers are also included in the Ulam spiral. The number 1 has only a single factor, itself; each prime number has two factors, itself and 1; composite numbers are divisible by at least three different factors. Using the size of the dot representing an integer to indicate the number of factors and coloring prime numbers red and composite numbers blue produces the figure shown.

Spirals following other tilings of the plane also generate lines rich in prime numbers, for example hexagonal spirals.

Klauber triangle with prime numbers generated by Euler's polynomial x² − x + 41 highlighted.
Sacks spiral.
Ulam spiral of size 150×150 showing both prime and composite numbers.
Hexagonal number spiral with prime numbers in green and more highly composite numbers in darker shades of blue.

References

1 2 3 Gardner 1964, p. 122.
↑ Stein, Ulam & Wells 1964, p. 520.
↑ Hoffman 1988, p. 41.
↑ Gardner 1971, p. 88.
↑ Hartwig, Daniel (2013), Guide to the Martin Gardner papers, The Online Archive of California, p. 117 .
↑ Weisstein, Eric W. "Prime Spiral". MathWorld.
↑ Jacobson Jr., M. J.; Williams, H. C (2003), "New quadratic polynomials with high densities of prime values", Mathematics of Computation, 72 (241): 499–519, doi:10.1090/S0025-5718-02-01418-7
↑ Guy, Richard K. (2004), Unsolved problems in number theory (3rd ed.), Springer, p. 8, ISBN 978-0-387-20860-2

Bibliography

Gardner, M. (March 1964), "Mathematical Games: The Remarkable Lore of the Prime Number", Scientific American, 210: 120–128, doi:10.1038/scientificamerican0364-120
Gardner, M. (1971), Martin Gardner's Sixth Book of Mathematical Diversions from Scientific American, University of Chicago Press, ISBN 978-0-226-28250-3
Hardy, G. H.; Littlewood, J. E. (1923), "Some Problems of 'Partitio Numerorum'; III: On the Expression of a Number as a Sum of Primes", Acta Mathematica, 44: 1–70, doi:10.1007/BF02403921
Hoffman, Paul (1988), Archimedes' Revenge: The Joys and Perils of Mathematics, New York: Fawcett Colombine, ISBN 0-449-00089-3
Stein, M. L.; Ulam, S. M.; Wells, M. B. (1964), "A Visual Display of Some Properties of the Distribution of Primes", American Mathematical Monthly, Mathematical Association of America, 71 (5): 516–520, doi:10.2307/2312588, JSTOR 2312588
Stein, M.; Ulam, S. M. (1967), "An Observation on the Distribution of Primes", American Mathematical Monthly, Mathematical Association of America, 74 (1): 43–44, doi:10.2307/2314055, JSTOR 2314055

External links

Wikimedia Commons has media related to Ulam spiral.

Prime Spirals - Numberphile, YouTube video with Dr. James Grime and the University of Nottingham
41 and more Ulam's Spiral - Numberphile, YouTube video with Dr. James Clewett and the University of Nottingham
Ulam spiral, Interactive Javascript application that displays the prime numbers up to 10¹⁸.

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