Raghib Abusaris and Omar Bayyati

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 24, 2018, Number 2, Pages 117—124

DOI: 10.7546/nntdm.2018.24.2.117-124

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## Details

### Authors and affiliations

Raghib Abusaris

*Department of Epidemiology and Biostatistics
College of Public Health and Health Informatics
King Saud bin Abdelaziz University for Health Science
Riyadh, Saudi Arabia
*

Omar Bayyati

*College of Sciences and Health Professions
King Saud bin Abdelaziz University for Health Science
Riyadh, Saudi Arabia
*

### Abstract

In this paper, we investigate the asymptotic behavior of the sequences generated by iterating the process of summing the modular powers of the decimal digits of a number. In particular, we identify all *modular happy numbers*. A number is called modular happy if the sequence obtained by iterating the process of summing the modular powers of the decimal digits of the number ends with 1.

### Keywords

- Happy numbers
- Sequences
- Recurrence relations
- Difference equations
- Modular arithmetic

### 2010 Mathematics Subject Classification

- 39A11

### References

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- Atanassov, K. (2015) A digital arithmetical function and some of its applications. Proceedings of the Jangjeon Mathematical Society, 18 (4), 511–528
- Guy, R. (2004) Unsolved problems in number theory (3rd ed.). Springer, New York.
- Rosen, K. H. (2012) Discrete Mathematics and Its Applications (7th ed.). McGraw-Hill, New York.
- Weisstein, E. W. (2017) Periodic sequence. Available online at http://mathworld.wolfram.com/PeriodicSequence.html. Retrieved March 9, 2017.

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## Cite this paper

APAAbusaris, R., & Bayyati, O. 2018). On modular happy numbers. Notes on Number Theory and Discrete Mathematics, 24(2), 117-124, doi: 10.7546/nntdm.2018.24.2.117-124.

ChicagoAbusaris, Raghib, and Omar Bayyati. “On Modular Happy Numbers.” Notes on Number Theory and Discrete Mathematics 24, no. 2 (2018): 117-124, doi: 10.7546/nntdm.2018.24.2.117-124.

MLAAbusaris, Raghib, and Omar Bayyati. “On Modular Happy Numbers.” Notes on Number Theory and Discrete Mathematics 24.2 (2018): 117-124. Print, doi: 10.7546/nntdm.2018.24.2.117-124.