Certain Continued Fraction Algorithm Converging Simultaneously under ℝ and ℚₚ (Analytic Number Theory and Related Topics)

Abstract

Let 𝘱 be a prime number and 𝘒 be a field with embedding into ℝ and ℚₚ. We propose an algorithm that generates continued fraction expansions converging in ℚₚ and is expected to simultaneously converge in both ℝ and ℚₚ. This algorithm produces finite continued fraction expansions for rational numbers. In cases where 𝘱 = 2, 3 and 𝘒 is a quadratic field, based on numerical experiments, we conjecture that the continued fraction expansions generated by this algorithm converge in both ℝ and ℚₚ. Furthermore, we anticipate that these expansions eventually exhibit periodicity or finiteness.