$p$-Adic quotient sets: diagonal forms

Abstract

For a set of integers $A$, we consider $R(A)=\{a/b:a,b\in A,b\ne 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of nonzero values attained by an integral form. This problem has been answered for quadratic forms. Very recently, Antony and Barman have studied this problem for the diagonal binary cubic forms $a{x}^3+b{y}^3$, where $a$ and $b$ are integers. In this article, we study this problem for diagonal forms. We extend the results of Antony and Barman to the diagonal binary forms $a{x}^n+b{y}^n$ for all $n\ge 3$. We also study $p$-adic denseness of quotients of nonzero values attained by diagonal forms of degree $n\ge 3$, where $\gcd (n,p(p-1))=1$.

8 pages. arXiv admin note: text overlap with arXiv:2110.05420