The Ulam spiral or prime spiral (in other languages also called the Ulam cloth) is a graphical depiction of the set of prime numbers, devised by mathematician Stanislaw Ulam in 1963 and popularized in Martin Gardner's Mathematical Games column in Scientific American a short time later. It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers.
Ulam and Gardner emphasized the striking appearance in the spiral of prominent diagonal, horizontal, and vertical lines containing large numbers of primes. Both Ulam and Gardner noted that the existence of such prominent lines is not unexpected, as lines in the spiral correspond to quadratic polynomials, and certain such polynomials, such as Euler's prime-generating polynomial x2 − x + 41, are believed to produce a high density of prime numbers. Nevertheless, the Ulam spiral is connected with major unsolved problems in number theory such as Landau's problems. In particular, no quadratic polynomial has ever been proved to generate infinitely many primes, much less to have a high asymptotic density of them, although there is a well-supported conjecture as to what that asymptotic density should be.
In 1932, more than thirty years prior to Ulam's discovery, the herpetologist Laurence M. Klauber constructed a triangular, non-spiral array containing vertical and diagonal lines exhibiting a similar concentration of prime numbers. Like Ulam, Klauber noted the connection with prime-generating polynomials, such as Euler's.
The Ulam spiral is produced by specially marking the prime numbers—for example by circling the primes or writing only the primes or by writing the prime numbers and non-prime numbers in different colors—to obtain a figure like the one below.
In the figure, primes appear to concentrate along certain diagonal lines. In the 200×200 Ulam spiral shown above, diagonal lines are clearly visible, confirming that the pattern continues. Horizontal and vertical lines with a high density of primes, while less prominent, are also evident. Most often, the number spiral is started with the number 1 at the center, but it is possible to start with any number, and the same concentration of primes along diagonal, horizontal, and vertical lines is observed. Starting with 41 at the center gives a particularly impressive example, with a diagonal containing an unbroken string of 40 primes, part of which is shown below.
According to Gardner, Ulam discovered the spiral in 1963 while doodling during the presentation of "a long and very boring paper" at a scientific meeting. These hand calculations amounted to "a few hundred points". Shortly afterwards, Ulam, with collaborators Myron Stein and Mark Wells, used MANIAC II at Los Alamos Scientific Laboratory to extend the calculation to about 100,000 points. The group also computed the density of primes among numbers up to 10,000,000 along some of the prime-rich lines as well as along some of the prime-poor lines. Images of the spiral up to 65,000 points were displayed on "a scope attached to the machine" and then photographed. The Ulam spiral was described in Martin Gardner's March 1964 Mathematical Games column in Scientific American and featured on the front cover of that issue. Some of the photographs of Stein, Ulam, and Wells were reproduced in the column.
In an addendum to the Scientific American column, Gardner mentioned the earlier paper of Klauber. Klauber describes his construction as follows, "The integers are arranged in triangular order with 1 at the apex, the second line containing numbers 2 to 4, the third 5 to 9, and so forth. When the primes have been indicated, it is found that there are concentrations in certain vertical and diagonal lines, and amongst these the so-called Euler sequences with high concentrations of primes are discovered."
Diagonal, horizontal, and vertical lines in the number spiral correspond to polynomials of the form
where b and c are integer constants. When b is even, the lines are diagonal, and either all numbers are odd, or all are even, depending on the value of c. It is therefore no surprise that all primes other than 2 line in alternate diagonals of the Ulam spiral. To understand why some odd diagonals have a higher concentration of primes than others, it is necessary to understand the behavior of the corresponding quadratic polynomials modulo odd primes.
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.
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
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 simpler, but obviously equivalent formula is given by:
, where runs over all primes, and — is number of zeros of the quadratic polynomials modulus 'p'.
It further simplifies to .
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, 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 is the Legendre symbol. A quadratic polynomial with A ≈ 11.3, currently the highest known value, has been discovered by Jacobson and Williams.
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.
- Gardner 1964, p. 122.
- Stein, Ulam & Wells 1964, p. 517.
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- Stein, Ulam & Wells 1964, p. 520.
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- Hartwig, Daniel (2013), Guide to the Martin Gardner papers, The Online Archive of California, p. 117.
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