On finding the smallest happy numbers of any heights

• Published in 2019
• Added on 2026-07-04

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.

Links

• https://arxiv.org/abs/1904.12032v2
• https://arxiv.org/pdf/1904.12032v2

Other information

BibTeX entry

@article{Onfindingthesmallesthappynumbersofanyheights,
	key = {Onfindingthesmallesthappynumbersofanyheights},
	type = {article},
	title = {On finding the smallest happy numbers of any heights},
	author = {Gabriel Lapointe},
	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.},
	comment = {},
	date_added = {2026-07-04},
	date_published = {2019-07-04},
	urls = {https://arxiv.org/abs/1904.12032v2,https://arxiv.org/pdf/1904.12032v2},
	collections = {integerology},
	url = {https://arxiv.org/abs/1904.12032v2 https://arxiv.org/pdf/1904.12032v2},
	year = 2019,
	urldate = {2026-07-04},
	archivePrefix = {arXiv},
	eprint = {1904.12032},
	primaryClass = {math.NT}
}