# On finding the smallest happy numbers of any heights

## TL;DR

This paper develops a recursive and algorithmic approach to identify the smallest happy numbers across different heights and bases, utilizing properties of height and modular arithmetic for efficient computation.

## Contribution

It introduces a recursive relationship and an algorithm for finding smallest happy numbers of any height in various bases, including binary and ternary, based on their properties.

## Key findings

- Derived recursive relationship between smallest happy numbers and height.
- Developed an algorithm exploiting height properties for computation.
- Established equations using modular arithmetic for binary and ternary bases.

## Abstract

This paper focuses on finding the smallest happy number for each height in any numerical base. Using the properties of the height, we deduce a recursive relationship between the smallest happy number and the height where the initial height is function of the numerical base. With the usage of the recursive relationship, we build an algorithm that exploits the properties of the height in order to find all of those smallest happy numbers with unknown height. However, with the modular arithmetic, we conclude on an equation that calculates the smallest happy numbers based on known heights for binary and ternary bases.