Arithmetical functions commutable with sums of squares

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
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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 k\in{\mathbb N}_0 and K\in \mathbb C, where {\mathbb N}_0, \mathbb C denote the set of nonnegative integers and complex numbers, respectively. We give all functions f, h_1, h_2, h_3, h_4:{\mathbb N}_0\to \mathbb C which satisfy the relation

    \[f(x_1^2+x_2^2+x_3^2+x_4^2+k)=h_1(x_1)+h_2(x_2)+h_3(x_3)+h_4(x_4)+K\]

for every x_1, x_2, x_3, x_4\in{\mathbb N}_0. We also give all arithmetical functions F, H_1, H_2, H_3, H_4:{\mathbb N}\to \mathbb C which satisfy the relation

    \[F(x_1^2+x_2^2+x_3^2+x_4^2+k)=H_1(x_1)+H_2(x_2)+H_3(x_3)+H_4(x_4)+K\]

for every x_1,x_2, x_3,x_4\in{\mathbb N}, where {\mathbb N} 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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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.

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