Sums of products of powers of given prime numbers
type: Item
instance of: scholarly article
title: Sums of products of powers of given prime numbers
zbMATH Open document ID: 0606.10012
author: Lianxiang Wang
author: R. Tijdeman
published in: Pacific Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: We give the complete solutions of the equations \[ 2^x 3^y+1=2^z+3^w,\quad 2^x 3^y+2^z=3^w+1\quad 2^x 3^y+3^w = 2^z+1 \quad\text{ in integers } x, y, z, w. \] We use this to prove that every large rational number has at most four representations of the form \(2^{\alpha} 3^{\beta}+2^{\gamma}+3^{\delta}\) with \(\alpha, \beta, \gamma, \delta\in\mathbb{Z}\). Finally we prove that, for given integer \(n\) and prime numbers \(p_1,\dots,p_t\) every rational number \(m\) has at most \(C\) representations of the form \(\sum^n_{i=1} p_i^{k_i}\) \((k_1,\dots,k_n\in\mathbb{Z})\) where \(C\) is an integer which depends only on \(n\), \(p_1,\dots,p_t\) and further an extension of this result to algebraic number fields. The proofs of these results are based on effective and ineffective methods from the theory of diophantine approximations. Note: By using a new result of Evertse and Győry the dependence of \(C_ 0\) on \(W\) in Theorem 6 can be relaxed.
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11D85
zbMATH DE Number: 3979150
zbMATH Keywords: exponential diophantine equations
zbMATH Keywords: representation of rationals
zbMATH Keywords: sums of power products
Wikidata QID: Q106102760
reviewed by: R. Tijdeman
MaRDI profile type: Publication
full work available at URL: pjm.1988.132.177
OpenAlex ID: W2048738986
DOI: 10.2140/PJM.1988.132.177
description: scientific article; zbMATH DE number 3979150