Completely $p$-primitive binary quadratic forms

Byeong-Kweon Oh; Hoseog Yu

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arXiv · 1705.04283

Completely $p$-primitive binary quadratic forms

Byeong-Kweon Oh·Hoseog Yu

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Abstract

Let $f(x,y)=ax^2+bxy+cy^2$ be a binary quadratic form with integer coefficients. For a prime $p$ not dividing the discriminant of $f$, we say $f$ is completely $p$-primitive if for any non-zero integer $N$, the diophantine equation $f(x,y)=N$ has always an integer solution $(x,y)=(m,n)$ with $(m,n,p)=1$ whenever it has an integer solution. In this article, we study various properties of completely $p$-primitive binary quadratic forms. In particular, we give a necessary and sufficient condition for a definite binary quadratic form $f$ to be completely $p$-primitive.

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BibTeX RIS

Byeong-Kweon Oh, Hoseog Yu. 2017-05-11. Completely $p$-primitive binary quadratic forms. https://arxiv.org/abs/1705.04283

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