Ulam Spiral - Hardy and Littlewood's Conjecture F

Hardy and Littlewood's Conjecture F

In their famous 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

In the first product, p is a 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.