Octahedrons with equally many lattice points and generalizations

Thomas Stoll*, Robert F. Tichy

*Corresponding author for this work

Research output: Contribution to conferencePaper

Abstract

While counting lattice points in octahedra of different dimensions n and m, it is an interesting question to ask, how many octahedra exist containing equally many such points. This gives rise to the Diophantine equation Pn(x) = P m(y) in rational integers x,y, where {P k(x)} denote special Meixner polynomials {M k , c)(x)} with β = 1, c = -1. We join the purely algebraic criterion of Y. Bilu and R. F. Tichy (The Diophantine equation f(x) = g(y), Acta Arith. 95 (2000), no. 3, 261-288) with a famous result of P. Erdös and J. L. Selfridge (The product of consecutive integers is never a power, Illinois J. Math. 19 (1975), 292-301) and prove that (euqation presented) with m, n ≥ 3, β ∈ ℤ/{0,-1,-2,-max(n,m) + 1} and c1,c 2 ∈ ℚ \ {0, 1} only admits a finite number of integral solutions x, y. Some more results on polynomial families in three-term recurrences are presented.

Original languageEnglish
Pages724-729
Number of pages6
Publication statusPublished - 1 Dec 2006
Event18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006 - San Diego, United States
Duration: 19 Jun 200623 Jun 2006

Conference

Conference18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006
CountryUnited States
CitySan Diego
Period19/06/0623/06/06

Keywords

  • Counting lattice points
  • Diophantine equations
  • Meixner polynomials
  • Orthogonal polynomials

ASJC Scopus subject areas

  • Algebra and Number Theory

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