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REPRESENTATIONS OF INTEGERS BY THE BINARY QUADRATIC FORM $x^{2}+xy+ny^{2}$

Published online by Cambridge University Press:  17 November 2015

BUMKYU CHO*
Affiliation:
Department of Mathematics, Dongguk University-Seoul, 30 Pildong-ro 1-gil, Jung-gu, Seoul100-715, Republic of Korea email bam@dongguk.edu
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Abstract

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In terms of class field theory we give a necessary and sufficient condition for an integer to be representable by the quadratic form $x^{2}+xy+ny^{2}$ ($n\in \mathbb{N}$ arbitrary) under extra conditions $x\equiv 1\;\text{mod}\;m$, $y\equiv 0\;\text{mod}\;m$ on the variables. We also give some examples where their extended ring class numbers are less than or equal to $3$.

Type
Research Article
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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