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
Equivalently, the ring of algebraic integers of
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 that the gap in Heegner's proof was minor.
which gives (distinct) primes for n = 0, ..., 39, is related to the Heegner number 163 = 4 · 41 − 1.
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.
[4] Ramanujan's constant is the transcendental number[5]
This number was discovered in 1859 by the mathematician Charles Hermite.
[7] In a 1975 April Fool article in Scientific American magazine,[8] "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.
In this wise it has as a spurious provenance as the Feynman point.
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.
and the minimal (monic integral) polynomial it satisfies is called the 'Hilbert class polynomial'.
The q-expansion of j, with its Fourier series expansion written as a Laurent series in terms of
and the low order coefficients grow more slowly than
For the four largest Heegner numbers, the approximations one obtains[9] are as follows.
where the reason for the squares is due to certain Eisenstein series.
[11] The integer j-invariants are highly factorisable, which follows from the form and factor as,
These transcendental numbers, in addition to being closely approximated by integers (which are simply algebraic numbers of degree 1), can be closely approximated by algebraic numbers of degree 3,[12]
The roots of the cubics can be exactly given by quotients of the Dedekind eta function η(τ), a modular function involving a 24th root, and which explains the 24 in the approximation.
They can also be closely approximated by algebraic numbers of degree 4,[13]
which, with the appropriate fractional power, are precisely the j-invariants.
Similarly for algebraic numbers of degree 6,
where the xs are given respectively by the appropriate root of the sextic equations,
These sextics are not only algebraic, they are also solvable in radicals as they factor into two cubics over the extension
These algebraic approximations can be exactly expressed in terms of Dedekind eta quotients.
where the eta quotients are the algebraic numbers given above.
The three numbers 88, 148, 232, for which the imaginary quadratic field
has class number 2, are not Heegner numbers but have certain similar properties in terms of almost integers.
[15] For details, see "Quadratic Polynomials Producing Consecutive Distinct Primes and Class Groups of Complex Quadratic Fields" by Richard Mollin.