Heegner number

In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field has class number 1. Equivalently, the ring of algebraic integers of has unique factorization. The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory. According to the (Baker–)Stark–Heegner theorem there are precisely nine Heegner numbers: This result was conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966, and Stark further indicated the gap in Heegner's proof was minor. Euler's prime-generating polynomial which gives (distinct) primes for n = 0, ..., 39, is related to the Heegner number 163 = 4 · 41 − 1. Rabinowitz proved that gives primes for if and only if this quadratic's discriminant is the negative of a Heegner number. (Note that yields , so is maximal.) 1, 2, and 3 are not of the required form, so the Heegner numbers that work are 7, 11, 19, 43, 67, 163, yielding prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais. Ramanujan's constant is the transcendental number which is an almost integer, in that it is very close to an integer: This number was discovered in 1859 by the mathematician Charles Hermite. In a 1975 April Fool article in Scientific American magazine, "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name. This coincidence is explained by complex multiplication and the q-expansion of the j-invariant. In what follows, j(z) denotes the j-invariant of the complex number z. Briefly, is an integer for d a Heegner number, and via the q-expansion. If is a quadratic irrational, then the j-invariant is an algebraic integer of degree , the class number of and the minimal (monic integral) polynomial it satisfies is called the 'Hilbert class polynomial'.