On finding the smallest happy numbers of any heights
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.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 DOI:
 10.48550/arXiv.1904.12032
 arXiv:
 arXiv:1904.12032
 Bibcode:
 2019arXiv190412032L
 Keywords:

 Mathematics  Number Theory;
 11B37;
 11Y55;
 11Y16;
 11A07
 EPrint:
 Dataset, unit tests and algorithms source code are available here: https://github.com/glapointe7/SmallestHappyNumbers