Positive binary forms representing the same integers in an arithmetic progression
Volume 181 / 2017
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 $k$.
