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Mat. Zametki, 2018, Volume 103, Issue 6, Pages 853–862 (Mi mz12006)  

Linear Congruences in Continued Fractions on Finite Alphabets

I. D. Kan

Moscow Aviation Institute (National Research University)

Abstract: A linear homogeneous congruence $ay\equiv bY  (\operatorname{mod}{q})$ is considered and an order-sharp upper bound for the number of its solutions is proved. Here $a$$b$, and $q$ are given jointly coprime numbers and $y$ and $Y$ are coprime variables in a given closed interval such that the number $y/Y$ can be expanded in a continued fraction with partial quotients from some alphabet $\mathbf{A}\subseteq\mathbb{N}$. For $\mathbf{A}=\mathbb{N}$ (and without the assumption that $y$ and $Y$ are coprime), a similar problem was solved by N. M. Korobov.

Keywords: linear congruence, continued fraction.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-05700 А
This work was supported by the Russian Foundation for Basic Research under grant 15-01-05700 A.


DOI: https://doi.org/10.4213/mzm12006

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English version:
Mathematical Notes, 2018, 103:6, 911–918

Bibliographic databases:

UDC: 511.321+511.31
Received: 18.05.2017
Revised: 14.07.2017

Citation: I. D. Kan, “Linear Congruences in Continued Fractions on Finite Alphabets”, Mat. Zametki, 103:6 (2018), 853–862; Math. Notes, 103:6 (2018), 911–918

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