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Multiplicative functions k-additive on hexagonal numbers

  • Elchin Hasanalizade EMAIL logo , Poo-Sung Park und Emil Inochkin
Veröffentlicht/Copyright: 24. Oktober 2025
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Abstract

Let k ≥ 3 be an integer. We prove that the set H of all nonzero hexagonal numbers is a k-additive uniqueness set for the set of multiplicative functions. That is, if a multiplicative function fk satisfies a multivariate Cauchy functional equation

f k ( x 1 + x 2 + + x k ) = f k ( x 1 ) + f k ( x 2 ) + + f k ( x k )

for arbitrary x1,,xkH , then fk is the identity function fk(n) = n for all nN . This extends the work of Kim et al. for k = 2.

Funding statement: This research was supported by ADA University Faculty Research and Development Funds and by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (NRF-2021R1A2C1092930)

Acknowledgement

The authors would like to thank the anonymous referee for his/her helpful comments.

  1. (Communicated by István Gaál)

References

[1] Chung, P. V.: Multiplicative functions satisfying the equation f(m2 + n2 = f(m2 + f(n2 Math. Slovaca 46 (1996), 165–171.Suche in Google Scholar

[2] Chung, P. V.—Phong, B. M.: Additive uniqueness sets for multiplicative functions, Publ. Math. Debrecen 55(3–4) (1999), 237–243.10.5486/PMD.1999.1950Suche in Google Scholar

[3] Dubickas, A.—Šarka, P.: On multiplicative functions which are additive on sums of primes, Aequationes Math. 86 (2013), 81–89.10.1007/s00010-012-0156-8Suche in Google Scholar

[4] Duke, W.—Schulze-Pillot, R.: Representations of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99 (1990), 49–57.10.1007/BF01234411Suche in Google Scholar

[5] Fang, J.-H.: A characterization of the identity function with equation f(p + q + r) = f(p) + f(q) + f(r), Combinatorica 31(6) (2011), 697–701.10.1007/s00493-011-2739-8Suche in Google Scholar

[6] Hasanalizade, E.: Multiplicative functions k-additive on generalized pentagonal numbers, Integers 22 (2022), #A43.Suche in Google Scholar

[7] Kim, B.—Kim, J. Y.—Lee, C. G.—Park, P.-S.: Multiplicative functions additive on generalized pentagonal numbers, C. R. Math. Acad. Sci. Paris 356(2) (2018), 125–128.10.1016/j.crma.2017.12.011Suche in Google Scholar

[8] Kim, B.—Kim, J. Y.—Lee, C. G.—Park, P.-S.:Multiplicative functions additive on polygonal numbers, Aequationes Math. 95 (2021), 601–621.10.1007/s00010-021-00824-8Suche in Google Scholar

[9] Legendre, A.-M.: Theorie des Nombres, A. Blanchard, Paris, 1979.Suche in Google Scholar

[10] Nathanson, M. B.: Additive Number Theory. The Classical Bases. Grad. Texts in Math. 164, Springer-Verlag, New York, 1996.10.1007/978-1-4757-3845-2Suche in Google Scholar

[11] Park, P.-S.: On k-additive uniqueness of the set of squares for multiplicative functions, Aequationes Math. 92(3) (2018), 487–495.10.1007/s00010-017-0517-4Suche in Google Scholar

[12] Park, P.-S.: Multiplicative functions which are additive on triangular numbers, Bull. Korean Math. Soc. 58(3) (2021), 603–608.Suche in Google Scholar

[13] Park, P.-S.: On multiplicative functions which are additive on positive cubes, Aequationes Math. 98(6) (2024), 1457–1474.10.1007/s00010-024-01118-5Suche in Google Scholar

[14] Spiro, C. A.: Additive uniqueness sets for arithmetic functions, J. Number Theory 42(2) (1992), 232–246.10.1016/0022-314X(92)90022-HSuche in Google Scholar

Received: 2024-12-31
Accepted: 2025-06-10
Published Online: 2025-10-24
Published in Print: 2025-10-27

© 2025 Mathematical Institute Slovak Academy of Sciences

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Heruntergeladen am 16.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2025-0074/html
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