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Peer-reviewedCC BY 4.0Axioms

Multiplicative Dependence of Shifted Quadratic and Cubic Algebraic Numbers

  • Paulius Drungilas1
  • Artūras Dubickas1
  1. 1Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania
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Abstract

We study short multiplicative relations among additive translates of algebraic numbers. Every quadratic algebraic number admits infinitely many relations with three factors on each side and arbitrarily large integer shifts. For quadratic algebraic integers, we classify the minimum total length over integer tails and show that it is eventually five. We also construct infinitely many reduced relations of length eight for every cubic algebraic number using rational shifts, and prove that the integer-shift length of 23 is exactly eight.

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Cite

BibTeXRIS
BibTeX
@article{drungilas2026multiplicative,
  title = {{Multiplicative Dependence of Shifted Quadratic and Cubic Algebraic Numbers}},
  author = {Paulius Drungilas and Artūras Dubickas},
  journal = {Axioms},
  year = {2026},
  month = oct,
  doi = {10.3390/axioms15100765},
  url = {https://www.mdpi.com/2075-1680/15/10/765},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Multiplicative Dependence of Shifted Quadratic and Cubic Algebraic Numbers
AU  - Paulius Drungilas
AU  - Artūras Dubickas
PY  - 2026
DA  - 2026/10/13
JO  - Axioms
DO  - 10.3390/axioms15100765
UR  - https://www.mdpi.com/2075-1680/15/10/765
AB  - We study short multiplicative relations among additive translates of algebraic numbers. Every quadratic algebraic number admits infinitely many relations with three factors on each side and arbitrarily large integer shifts. For quadratic algebraic integers, we classify the minimum total length over integer tails and show that it is eventually five. We also construct infinitely many reduced relations of length eight for every cubic algebraic number using rational shifts, and prove that the integer-shift length of 23 is exactly eight.
SN  - 2075-1680
ER  -