Congruences mod 4 for the alternating sum of the partial quotients

TL;DR

This paper derives congruences modulo 4 for the alternating sum of partial quotients in continued fractions of rational numbers, using techniques from Dedekind sums and the Barkan-Hickerson-Knuth formula.

Contribution

It introduces a novel method combining Dedekind sum techniques with the Barkan-Hickerson-Knuth formula to analyze partial quotients.

Findings

Established congruences mod 4 for the alternating sum of partial quotients.

Connected continued fraction properties with Dedekind sum congruences.

Provided a new approach to study continued fractions via number-theoretic techniques.

Abstract

We apply a technique used in $[$Tsukerman, Equality of Dedekind sums mod $\mathbb Z$, $2\mathbb Z$ and $4\mathbb Z$, arXiv:1408.3225] combined with the Barkan-Hickerson-Knuth-formula in order to obtain congruences mod $4$ for the alternating sum of the partial quotients of the continued fraction expansion of $a/b$, where $0<a<b$ are integers.