Linear Inhomogeneous Congruences in Continued Fractions on Finite Alphabets

We consider the linear inhomogeneous congruence and prove an upper estimate for the number of its solutions. Here , , , and are given natural numbers, and are coprime variables from a given interval such that the number expands in a continued fraction with partial quotients on a finite alphabet . Fo...

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Bibliographic Details
Published in:Mathematical Notes 2022-10, Vol.112 (3-4), p.424-435
Main Authors: Kan, I. D., Odnorob, V. A.
Format: Article
Language:English
Subjects:
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639/766/189
639/766/530
639/766/747
Congruences
Fractions
Linear functions
Mathematics
Mathematics and Statistics
Number theory
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Summary:We consider the linear inhomogeneous congruence and prove an upper estimate for the number of its solutions. Here , , , and are given natural numbers, and are coprime variables from a given interval such that the number expands in a continued fraction with partial quotients on a finite alphabet . For , a similar problem has been solved earlier by I. D. Kan and, for , by N. M. Korobov. In addition, in one of the recent statements of the problem, an additional constraint in the form of a linear inequality was also imposed on the fraction .
ISSN:0001-4346
1067-9073
1573-8876
DOI:10.1134/S0001434622090115