Octahedrons with equally many lattice points (Q5933699)

For positive integers \(n,r\) denote by \(p_n(r)\) the number of points \((x_1,\hdots,x_n)\in {\mathbb Z}^n\) satisfying \(|x_1|+\hdots+|x_n|\leq r\). \textit{L. Hajdu} [Acta Math. Hung. 78, No. 1-2, 59-70 (1998; Zbl 0903.11011); Publ. Math. 51, No. 3-4, 331-342 (1997; Zbl 0903.11012)] studied the equation \(p_n(x)=p_m(y)\) for small \(n\) and \(m\). He conjectured that the equation has finitely many solutions when \(n>m=2\). This conjecture was confirmed by \textit{P. Kirschenhofer}, \textit{A. Pethö} and \textit{R. F. Tichy} [Acta Sci. Math. 65, No. 1-2, 47-59 (1999; Zbl 0983.11013)]. In the present paper the authors extend this result to the general case by showing that the above equation has at most finitely many solutions in integers \(x,y\) for any pair \((n,m)\) of distinct positive integers with \(n,m\geq 2\). The main tool of the proof is a recent powerful result of \textit{Y. F. Bilu} and \textit{R. F. Tichy} on Diophantine equations of the form \(f(x)=g(y)\) where \(f,g\in {\mathbb Z}[x]\), see [Acta Arith. 95, No. 3, 261-288 (2000; Zbl 0958.11049)].