Some divisibility properties of binomial coefficients
| Autor: | Madjid Mirzavaziri, Daniel Yaqubi |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Discrete mathematics
Algebra and Number Theory Mathematics::Commutative Algebra Coprime integers Gaussian integer 010102 general mathematics Euler's totient function Lucky numbers of Euler 010103 numerical & computational mathematics Composition (combinatorics) 01 natural sciences Combinatorics symbols.namesake Quadratic integer Eisenstein integer FOS: Mathematics symbols Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics Binomial coefficient Mathematics |
| Zdroj: | Journal of Number Theory. 183:428-441 |
| ISSN: | 0022-314X |
| DOI: | 10.1016/j.jnt.2017.08.005 |
| Popis: | In this paper, we aim to give full or partial proofs for the following three conjectures of V. J. W. Guo and C. Krattenthaler: (1) Let a > b be positive integers, α , β be any integers and p be a prime satisfying gcd ( p , a ) = 1 . Then there exist infinitely many positive integers n for which ( a n + α b n + β ) ≡ r ( mod p ) for all integers r ; (2) For any odd prime p , there are no positive integers a > b such that ( a n b n ) ≡ 0 ( mod p n − 1 ) for all n ≥ 1 ; (3) For any positive integer m , there exist positive integers a and b such that a m > b and ( a m n b n ) ≡ 0 ( mod a n − 1 ) for all n ≥ 1 . |
| Databáze: | OpenAIRE |
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