Heegner Number - Euler's Prime-generating Polynomial

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 its discriminant equals minus 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, yielding prime generating functions of Euler's form for ; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.