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单词 Heegner Number
释义

Heegner Number

The values of for which Quadratic Fields are uniquely factorable intofactors of the form . Here, and are half-integers, except for and 2, in which case they areIntegers. The Heegner numbers therefore correspond to Discriminants which have Class Number equal to 1, except for Heegner numbers and , which correspond to and, respectively.


The determination of these numbers is called Gauss's Class Number Problem, and it is now knownthat there are only nine Heegner numbers: , , , , , , , , and (Sloane's A003173),corresponding to discriminants , , , , , , , , and , respectively.


Heilbronn and Linfoot (1934) showed that if a larger existed, it must be . Heegner (1952) published a proofthat only nine such numbers exist, but his proof was not accepted as complete at the time. Subsequentexamination of Heegner's proof show it to be ``essentially'' correct (Conway and Guy 1996).


The Heegner numbers have a number of fascinating connections with amazing results in Prime Number theory.In particular, the j-Function provides stunning connections between , , and the AlgebraicIntegers. They also explain why Euler's Prime-Generating Polynomial is sosurprisingly good at producing Primes.

See also Class Number, Discriminant (Binary Quadratic Form), Gauss's Class Number Problem,j-Function, Prime-Generating Polynomial, Quadratic Field


References

Conway, J. H. and Guy, R. K. ``The Nine Magic Discriminants.'' In The Book of Numbers. New York: Springer-Verlag, pp. 224-226, 1996.

Heegner, K. ``Diophantische Analysis und Modulfunktionen.'' Math. Z. 56, 227-253, 1952.

Heilbronn, H. A. and Linfoot, E. H. ``On the Imaginary Quadratic Corpora of Class-Number One.'' Quart. J. Math. (Oxford) 5, 293-301, 1934.

Sloane, N. J. A. SequenceA003173/M0827in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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