Happy Numbers and Fixed Points

Authors

  • Rudney Carlos da Mata Instituto Federal de Educação, Ciência e Tecnologia de Minas Gerais (IFMG), Ouro Branco, MG, Brasil https://orcid.org/0000-0003-3549-2490
  • Marcelo Oliveira Veloso Universidade Federal de São João del-Rei (UFSJ), Departamento de Estatística, Física e Matemática, Ouro Branco, MG, Brasil https://orcid.org/0000-0002-2585-1210

DOI:

https://doi.org/10.35819/remat2023v9i1id6190

Keywords:

Happy Numbers, Fixed Points, Sequence of Integers

Abstract

This paper presents a brief study about the set of happy numbers, in any positional basis b>=2. We show examples of happy numbers and verify that every positive integer is happy in basis 4, Example 2.8. In particular, we characterize the fixed points of the happiness function, Theorem 3.2, which assigns to every positive integer the sum of the squares of its digits. In addition, techniques to determine the fixed points of the happiness function are showed, Theorem 3.5, Examples 3.7 and 3.8, in any positional basis.

Downloads

Download data is not yet available.

Author Biographies

Rudney Carlos da Mata, Instituto Federal de Educação, Ciência e Tecnologia de Minas Gerais (IFMG), Ouro Branco, MG, Brasil

http://lattes.cnpq.br/6500455969704457

Marcelo Oliveira Veloso, Universidade Federal de São João del-Rei (UFSJ), Departamento de Estatística, Física e Matemática, Ouro Branco, MG, Brasil

http://lattes.cnpq.br/4008756490618057

References

BEARDON, A. F. Sums of squares of digits. The Mathematical Gazette. v. 82, p. 379-388, 1998.

EL-SEDY, E.; SIKSEK, S. On Happy Numbers. Rocky Mountain Journal of Mathematics. v. 30, n. 2, p. 565-570, 2000. DOI: www.doi.org/10.1216/rmjm/1022009281.

GARCIA, A.; LEQUAIN, Yves. Elementos de Álgebra. 6. ed. Rio de Janeiro: IMPA, 2018.

GRUDMAN, H. G.; TEEPLE, E. A. Generalized Happy Numbers. The Fibonacci Quarterly. p. 462-466, 2001. Disponível em: https://www.fq.math.ca/Scanned/39-5/grundman.pdf. Acesso em: 22 jan. 2023.

GUY, R. Unsolved Problems in Number Theory. 3. ed. New York: Springer-Verlag, 2004.

HEFEZ, A. Aritmética. 2. ed. Rio de Janeiro: Coleção PROFMAT, SBM, 2016.

MUTTER, S. A. Happy Numbers: An Exploration of An Iterated Function in Different Bases. A Senior Project submitted to The Division of Science, Mathematics, and Computing of Bard College, 2010. Disponível em: https://media.gradebuddy.com/documents/1849233/5193b459-c8ea-46a9-a583-580895ca1a7e.pdf. Acesso em: 21 fev. 2023.

PAN, H. On consecutive happy numbers. Journal of Number Theory. v. 128, n. 6, p. 1646-1654, jun. 2008. DOI: www.doi.org/10.1016/j.jnt.2007.11.009.

Downloads

Published

2023-03-07

How to Cite

MATA, R. C. da; VELOSO, M. O. Happy Numbers and Fixed Points. REMAT: Revista Eletrônica da Matemática, Bento Gonçalves, RS, v. 9, n. 1, p. e3002, 2023. DOI: 10.35819/remat2023v9i1id6190. Disponível em: https://periodicos.ifrs.edu.br/index.php/REMAT/article/view/6190. Acesso em: 27 jul. 2024.

Issue

Vol. 9 No. 1 (2023)

Section

Mathematics

License

Copyright (c) 2023 REMAT: Revista Eletrônica da Matemática

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

REMAT retains the copyright of published articles, having the right to first publication of the work, mention of first publication in the journal in other published media and distribution of parts or of the work as a whole in order to promote the magazine.

This is an open access journal, which means that all content is available free of charge, at no cost to the user or his institution. Users are permitted to read, download, copy, distribute, print, search or link the full texts of the articles, or use them for any other legal purpose, without requesting prior permission from the magazine or the author. This statement is in accordance with the BOAI definition of open access.