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Counting consecutive multiplicatively dependent triples
Abstract
We obtain a nontrivial bound on the number of triples of positive integers (a,b,c) ∈ [1,H]3, which are multiplicatively dependent but each pair of its elements is not, and so is the triple $ (a+1,b+1,c+1)$. The method is based on studying factorisations of some resultants and a recent improvement by G. Binyamini, R. Cluckers and F. Kato (2025) of the classical Bombieri--Pila bound.
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BibTeX
@misc{shparlinski2026counting,
title = {{Counting consecutive multiplicatively dependent triples}},
author = {Igor Shparlinski and Nicolas Sleiman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29408},
eprint = {2609.29408},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29408},
}
RIS
TY - UNPB
TI - Counting consecutive multiplicatively dependent triples
AU - Igor Shparlinski
AU - Nicolas Sleiman
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29408
UR - https://arxiv.org/abs/2609.29408
AB - We obtain a nontrivial bound on the number of triples of positive integers (a,b,c) ∈ [1,H]^3, which are multiplicatively dependent but each pair of its elements is not, and so is the triple $ (a+1,b+1,c+1)$. The method is based on studying factorisations of some resultants and a recent improvement by G. Binyamini, R. Cluckers and F. Kato (2025) of the classical Bombieri--Pila bound.
ER -