Positive binary forms representing the same integers in an arithmetic progression

Myung-Hwan Kim ¹   ; Byeong-Kweon Oh ²

¹ Department of Mathematical Sciences Seoul National University Seoul 08826, Korea
² Department of Mathematical Sciences and Research Institute of Mathematics Seoul National University Seoul 08826, Korea

Acta Arithmetica, Tome 181 (2017) no. 2, pp. 111-126

Résumé

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 $k$.

Détail

DOI : 10.4064/aa8443-6-2017

Keywords: delone proved unique pair non equivalent positive definite primitive integral binary forms representing integers provide effective criteria finding pairs positive definite integral binary forms representing integers set prime non negative integer where set containing arithmetic progression common difference initial term nbsp