Octahedrons with equally many lattice points and generalizations

Thomas Stoll; Robert F. Tichy

Octahedrons with equally many lattice points and generalizations

^*Corresponding author for this work

Research output: Contribution to conference › Paper › peer-review

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 ^(β, ^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 language	English
Pages	724-729
Number of pages	6
Publication status	Published - 1 Dec 2006
Event	18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006 - San Diego, United States Duration: 19 Jun 2006 → 23 Jun 2006

Conference

Conference	18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006
Country/Territory	United States
City	San Diego
Period	19/06/06 → 23/06/06

Keywords

Counting lattice points
Diophantine equations
Meixner polynomials
Orthogonal polynomials

ASJC Scopus subject areas

Algebra and Number Theory

Cite this

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title = "Octahedrons with equally many lattice points and generalizations",

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 (β, 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. Erd{\"o}s 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.",

keywords = "Counting lattice points, Diophantine equations, Meixner polynomials, Orthogonal polynomials",

author = "Thomas Stoll and Tichy, {Robert F.}",

year = "2006",

month = dec,

day = "1",

language = "English",

pages = "724--729",

note = "18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006 ; Conference date: 19-06-2006 Through 23-06-2006",

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T1 - Octahedrons with equally many lattice points and generalizations

AU - Stoll, Thomas

AU - Tichy, Robert F.

PY - 2006/12/1

Y1 - 2006/12/1

N2 - 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 (β, 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.

AB - 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 (β, 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.

KW - Counting lattice points

KW - Diophantine equations

KW - Meixner polynomials

KW - Orthogonal polynomials

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M3 - Paper

AN - SCOPUS:84860622699

SP - 724

EP - 729

T2 - 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006

Y2 - 19 June 2006 through 23 June 2006

ER -

Octahedrons with equally many lattice points and generalizations

Abstract

Conference

Keywords

ASJC Scopus subject areas

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