On the representation of integers by \(p\)-adic diagonal forms
type: Item
instance of: scholarly article
title: On the representation of integers by \(p\)-adic diagonal forms
zbMATH Open document ID: 0434.10015
DOI: 10.1016/0022-314X(80)90029-3
author: Edie Stevenson
published in: Journal of Number Theory
publication date: 1980-01-01T00:00:00Z
review text: Let \(p\) be an odd prime, let \(d\) be a positive integer such that \((d,p-1)=1\), let \(r\) denote the \(p\)-adic valuation of \(d\) and let \(m=1+3+3^2+\ldots+3^r\). It is shown that for every \(p\)-adic integer \(n\) the equation \(\sum_{i=1}^m X_i^d=n\) has a nontrivial \(p\)-adic solution. It is also shown that for all \(p\)-adic units \(a_1, a_2, a_3, a_4\) and all \(p\)-adic integers \(n\) the equation \(\sum_{i=1}^4 a_iX_i^p=n\) has a nontrivial \(p\)-adic solution. A corollary to each of these results is that every \(p\)-adic integer is a sum of four \(p\)th powers of \(p\)-adic integers.
Mathematics Subject Classification ID: 11D88
Mathematics Subject Classification ID: 11D85
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 11E95
zbMATH DE Number: 3675992
zbMATH Keywords: representation of integers
zbMATH Keywords: p-adic diagonal forms
zbMATH Keywords: p-adic valuation
zbMATH Keywords: p-adic integer
MaRDI profile type: Publication
cites work: Q4652648
full work available at URL: 0022-314x(80)90029-3
OpenAlex ID: W2031936449
description: scientific article; zbMATH DE number 3675992