On a diophantine equation concerning the number of integer points in special domains
type: Item
instance of: scholarly article
title: On a diophantine equation concerning the number of integer points in special domains
zbMATH Open document ID: 0903.11011
DOI: 10.1023/A:1006518403429
published in: Acta Mathematica Hungarica
publication date: 1998-11-24T00:00:00Z
review text: Let \(f_n(r) = \# \{ (x_1, \dots ,x_n) \in {\mathbb Z}^n : | x_1| + \dots +| x_n| \leq r \}. \) Then \(f_n(r)\) is a polynomial with rational coefficients of degree \(n\). It is proved in this paper that the diophantine equations \[ f_2(r) = f_3(R)\quad\text{ and }\quad f_2(r) = f_4(R) \] have the solutions \((r,R)=(0,0),(3,2),(403,62),(6012,378)\) and \((r,R)=(0,0),(4,2)\), respectively. The proof of the first statement is based on the method of \textit{J. Gebel, H. G. Zimmer} and \textit{A. Pethő} [Acta Arith. 68, 171-192 (1994; Zbl 0816.11019)] and uses the computer algebra system SIMATH. To prove the second statement, the author applies a classical method of \textit{L. J. Mordell} [Pac. J. Math. 13, 1347-1351 (1963; Zbl 0124.27402)]. The author conjectures that the equation \(f_2(r)=f_n(R), n \geq 0, n\not= 2\) has only a finite number of solutions in non-negative integers \(r\) and \(R.\) As \(f_1(R)=2R+1\), this conjecture is false. Assuming however \(n \geq 3\) it becomes true, as will be shown in a forthcoming paper of \textit{P. Kirschenhofer, R. F. Tichy} and the reviewer. For Part II see the review Zbl 0903.11012 below.
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11P21
zbMATH DE Number: 1226457
zbMATH Keywords: elliptic equations
zbMATH Keywords: quartic diophantine equations
author: Lajos Hajdu
reviewed by: Attila Pethő
describes a project that uses: SIMATH
MaRDI profile type: Publication
full work available at URL: a:1006518403429
OpenAlex ID: W1515000379
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description: scientific article; zbMATH DE number 1226457