Heegner number

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In number theory, Heegner numbers are square-free positive integers d {\displaystyle d} such that the imaginary quadratic field Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} has class number 1. Equivalently, the ring of algebraic integers of Z [ − d ] {\displaystyle \mathbb {Z} [{\sqrt {-d}}]} 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 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...

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