Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ARITHMETIC FUNCTIONS COMMUTABLE WITH SUMS OF POWERS AND DIAGONAL QUADRATIC FORMS

  • Chimoon An (Department of Mathematics Ajou University) ;
  • Yongjin Jeon (Department of Mathematics Ajou University) ;
  • Chanyeong Lee (Department of Mathematics Ajou University) ;
  • Jungin Lee (Department of Mathematics Ajou University)
  • Received : 2025.07.15
  • Accepted : 2025.10.02
  • Published : 2026.07.31
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Abstract

Given a polynomial g(x1, … , xk) with integer coefficients, we say a function f : ℕ → ℂ is g-commutable if f(g(x1, … , xk)) = g(f(x1), … , f(xk)) for every x1, … , xk ∈ ℕ. In this paper, we characterize all g-commutable functions where g is one of the following: (1) g(x1, … , xk) = x31 + … + x3k for some positive integer k ≥ 3; (2) g(x, y) = Ax2 + By2 for some positive integers A and B.

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Acknowledgement

Jungin Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2024-00334558 and No. RS-2025-02262988) and the new faculty research fund of Ajou University (S-2023-G0001-00236). Chimoon An, Yongjin Jeon and Chanyeong Lee were partially supported by the new faculty research fund of Ajou University (S-2023-G0001-00236). We thank the anonymous referee for helpful comments that improved the exposition of the paper.