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Ternary quadratic forms $ax^2+by^2+cz^2$ representing all positive integers $8k+4$

Volume 166 / 2014

Kenneth S. Williams Acta Arithmetica 166 (2014), 391-396 MSC: Primary 11E20; Secondary 11E25. DOI: 10.4064/aa166-4-4

Abstract

Under the assumption that the ternary form $x^2+2y^2+5z^2+xz$ represents all odd positive integers, we prove that a ternary quadratic form $ax^2+by^2+cz^2$ $(a,b,c \in \mathbb {N})$ represents all positive integers $n\equiv 4\ ({\rm mod}\ 8)$ if and only if it represents the eight integers $4,12,20,28,52,$ $60,140$ and $308$.

Authors

  • Kenneth S. WilliamsSchool of Mathematics and Statistics
    Carleton University
    Ottawa, Ontario, Canada K1S 5B6
    e-mail

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