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Positive binary forms representing the same integers in an arithmetic progression

Volume 181 / 2017

Myung-Hwan Kim, Byeong-Kweon Oh Acta Arithmetica 181 (2017), 111-126 MSC: Primary 11E16; Secondary 11E12. DOI: 10.4064/aa8443-6-2017 Published online: 27 October 2017

Abstract

In 1938, Delone proved that $(x^2+3y^2,x^2+xy+y^2)$ is the unique pair of non-equivalent positive definite primitive integral binary forms representing the same integers. We provide effective criteria on finding all pairs of positive definite integral binary forms representing the same integers in the set $A_{p,k}$ for any prime $p$ and any non-negative integer $k$ less than $p$, where $A_{p,k}$ is the set containing an arithmetic progression with common difference $p$ and initial term $k$.

Authors

  • Myung-Hwan KimDepartment of Mathematical Sciences
    Seoul National University
    Seoul 08826, Korea
    e-mail
  • Byeong-Kweon OhDepartment of Mathematical Sciences and
    Research Institute of Mathematics
    Seoul National University
    Seoul 08826, Korea
    e-mail

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