I. Kátai and B. M. Phong

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 3, Pages 143–154

DOI: 10.7546/nntdm.2021.27.3.143-154

**Full paper (PDF, 177 Kb)**

## Details

### Authors and affiliations

I. Kátai

*Department of Computer Algebra, University of Eötvös Loránd
1117 Budapest, Hungary*

B. M. Phong

*Department of Computer Algebra, University of Eötvös Loránd
1117 Budapest, Hungary*

### Abstract

Let and , where , denote the set of nonnegative integers and complex numbers, respectively. We give all functions which satisfy the relation

for every . We also give all arithmetical functions which satisfy the relation

for every , where denotes the set of all positive integers.

### Keywords

- Arithmetical function
- Function equation
- Sums of squares
- Lagrange’s Four-Square Theorem

### 2020 Mathematics Subject Classification

- 11K65
- 11N37
- 11N64

### References

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*f*(*m*^{2}+*n*^{2}) =*f*(*m*^{2})+*f*(*n*^{2}). Mathematica Slovaca, 46, 165–171. - Kátai, I., & Phong, B. M. M. (2021). A characterization of functions using Lagrange’s Four-Square Theorem. Annales Universitatis Scientiarium Budapestinensis de Rolando Eötvös Nominatae. Sectio Computatorica, 52. (accepted)
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*f*(*n*^{2}+*Dm*^{2}+*k*) =*f*(*n*)^{2}+*Df*(*m*)^{2}+*k*, Annales Universitatis Scientiarium Budapestinensis de Rolando Eötvös Nominatae. Sectio Computatorica, 52. (accepted) - Park, P.-S. (2018). Multiplicative functions commutable with sums of squares. International Journal of Number Theory, 2, 469–478.
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## Cite this paper

Kátai, I., & Phong, B. M. (2021). Arithmetical functions commutable with sums of squares. *Notes on Number Theory and Discrete Mathematics*, 27(3), 143-154, DOI: 10.7546/nntdm.2021.27.3.143-154.