On Prielipp's problem on signed sums of \(k\)th powers
type: Item
instance of: scholarly article
title: On Prielipp's problem on signed sums of \(k\)th powers
zbMATH Open document ID: 0841.11002
author: Michael N. Bleicher
published in: Journal of Number Theory
publication date: 1996-03-19T00:00:00Z
review text: Starting with a question of R. W. Prielipp (concerning the special case \(k=2\) of the author's main result), the author proves by elementary (number-theoretical and combinatorial) methods: For every positive integer \(n\) and any integer \(k\geq 0\), there is an integer \(m\) and a vector \(\{\varepsilon_0, \dots, \varepsilon_m\}\), where every \(\varepsilon_i\) is from the set \(\{+1, -1\}\), such that \((**)\) \(n= \sum^m_{i=0} \varepsilon_i \cdot i^k\). For the ``length'' \(L(n)\) of this representation the author gives a lower and an upper bound, which -- for fixed \(k\) -- implies for the minimal \(m\) in \((**)\) the asymptotic relation, as \(n\to \infty\): \[ L(n)\sim [( k+1)\cdot n]^{1\over {k+1}}. \] Similarly, for fixed \(n\), a lower bound \(\ell(k)\geq k+2\), as \(k\to \infty\), is given. Finally, some problems and conjectures are mentioned.
Mathematics Subject Classification ID: 11A67
Mathematics Subject Classification ID: 11D68
Mathematics Subject Classification ID: 11D79
zbMATH DE Number: 844639
zbMATH Keywords: congruences of the \(k\)th power sums
zbMATH Keywords: signed sums of higher powers
zbMATH Keywords: representation of integers
zbMATH Keywords: representations as sums and differences
zbMATH Keywords: congruences
zbMATH Keywords: problems
zbMATH Keywords: conjectures
reviewed by: Wolfgang Schwarz
MaRDI profile type: Publication
full work available at URL: jnth.1996.0004
OpenAlex ID: W2065495804
DOI: 10.1006/JNTH.1996.0004
description: scientific article; zbMATH DE number 844639