Fixed Points of Augmented Generalized Happy Functions

TL;DR

This paper investigates fixed points of augmented generalized happy functions, analyzing their properties, counting fixed points for various parameters, and demonstrating the existence of parameter ranges with no fixed points.

Contribution

It provides new insights into the fixed points of augmented generalized happy functions, including counting and existence results across different parameters.

Findings

Counted fixed points for $S_{[c,b]}$ with $b \\geq 2$ and $0<c<3b-3$.

Proved existence of arbitrarily many $c$ values with no fixed points for each base $b \\geq 2$.

Abstract

An augmented generalized happy function $S_{[c,b]}$ maps a positive integer to the sum of the squares of its base $b$ digits plus $c$. In this paper, we study various properties of the fixed points of $S_{[c,b]}$; count the number of fixed points of $\S_{[c,b]}$, for $b \geq 2$ and $0<c<3b-3$; and prove that, for each $b \geq 2$, there exist arbitrarily many consecutive values of $c$ for which $S_{[c,b]}$ has no fixed point.