On algebraic independence of three p-adic continued fractions

Sarra Ahallal; Mohamed Begare; Ali Kacha

On algebraic independence of three p-adic continued fractions

Number Theory 2026-04-03 v1

Authors: Sarra Ahallal , Mohamed Begare , Ali Kacha

Abstract

In this paper, we establish sufficient conditions on the elements of the p-adic continued fractions $A$ and $B$ which guarantee that the p-adic continued fractions $A, B$ and $A^{B}$ are algebraically independent over $\mathbb{Q}$ . These elements have partial quotients that increase rapidly. We note that these results extend some work of Bundschuh. Furthermore, we give some numerical examples which illustrated the theoretical results.

Keywords

p-adic numbers continued fractions algebraic independence transcendental numbers Diophantine approximation

Cite

@article{arxiv.2604.01464,
  title  = {On algebraic independence of three p-adic continued fractions},
  author = {Sarra Ahallal and Mohamed Begare and Ali Kacha},
  journal= {arXiv preprint arXiv:2604.01464},
  year   = {2026}
}

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