On a diophantine equation concerning the number of integer points in special domains
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.
scientific article; zbMATH DE number 1226457