# Continuum cardinality of the set of solutions of one class of equations that contain the function of frequency of ternary digits of a number

### Abstract

We study the equation*v*

_{1 }(

*x*) =

*x*, where

*v*

_{1 }(

*x*) is the function of frequency of the digit 1 in ternary expansion of

*x*. We prove that this equation has a unique rational solution and a continuum set of irrational solutions. An algorithm for the construction of solutions is proposed. We also describe the topological and metric properties of the set of all solutions. Some additional facts about equations

*v*(

_{i }*x*),

*i*= 0,2, are also given.

Published

25.10.2008

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 60, no. 10, Oct. 2008, pp. 1414–1421, https://umj.imath.kiev.ua/index.php/umj/article/view/3255.

Issue

Section

Short communications