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 Q(√−d) has class number 1. Equivalently, its ring of integers 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 = 1, ..., 40, is related to the Heegner number 163 = 4 · 41 − 1.
Euler's formula, with taking the values 1,... 40 is equivalent to
with taking the values 0,... 39, and Rabinowitz proved that
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