On algebraic independence of three p-adic continued fractions

TL;DR

This paper provides conditions under which three p-adic continued fractions, including an element and its power, are algebraically independent over the rationals, extending previous work and supported by numerical examples.

Contribution

It establishes new sufficient conditions for algebraic independence of p-adic continued fractions and their powers, extending prior research in the area.

Findings

Conditions on partial quotients ensure algebraic independence

Results extend previous work of Bundschuh

Numerical examples support theoretical findings

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.