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In number theory, a Heegner number is a square-free positive integer ''d'' such that the imaginary quadratic field Q() 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 Stark–Heegner theorem there are precisely nine Heegner numbers: :, , , , , , , , . This result was conjectured by Gauss and proven by Kurt Heegner in 1952. ==Euler's prime-generating polynomial== 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〔Rabinowitz, G. "Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern." Proc. Fifth Internat. Congress Math. (Cambridge) 1, 418–421, 1913.〕 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.〔Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 88 and 144, 1983.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Heegner number」の詳細全文を読む スポンサード リンク
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