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Date Published:
27/02/2023
Online Published:
28/02/2023
Abstract Views: 63
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Views PDF: 60
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Kátai, I. ., B, K., & B, P. (2023). ON THE EQUATION F(n² + m² + k) = H(n) + H(m) + K. Thang Long Journal of Science: Mathematics and Mathematical Sciences, 1(4), 135-151. https://science.thanglong.edu.vn/index.php/volc/article/view/54
ON THE EQUATION F(n² + m² + k) = H(n) + H(m) + K
Main Article Content
Abstract
We give all solutions of the equation
F(n2 + m2 + k) = H(n) + H(m) + K (∀n;m ∈ N),
where k ∈ N is the sum of two xed squares, K ∈ C and F, H are completely multiplicative functions.
Article Details
Keywords
Arithmetical function, equation of functions
References
[1] Bojan Basic, Characterization of arithmetic functions that preserve the sum-of-squares operation, Acta Mathematica Sinica, English Series, 30 (2014), Issue 4, pp 689-695.
[2] Chung P. V, Multiplicative functions satisfying the equation f(n² + m²) = f(n²) + f(m²), Mathematica Slovaca, 46 (1996), 165-171.
[3] Fehér J. and I. K atai, Sets of uniqueness for additive and multiplicative functions, Ann. Univ. Sci. Budapest. Sect. Math., 47 (2002), 3-16.
[4] Fehér J., K.-H. Indlekofer and N. M. Timofeev, A set of uniqueness for completely additive arithmetic functions, Annales Univ. Sci. Budapest. Sect. Comp., 21(2004), 57-67.
[5] I. Kátai and B. M. Phong, The functional equation f(A+B) = g(A)+ h(B), Annales Univ. Sci. Budapest. Sect. Comp., 43 (2014), 287-301.
[6] Kátai I. and B. M. Phong, Some unsolved problems on arithmetical functions, Ann. Univ. Sci. Budapest. Sect. Comput., 44 (2015), 233-235.
[7] Kátai I. and B. M. Phong, A characterization of functions using Lagrange's Four-Square Theorem, Annales Univ. Sci. Budapest., Sect. Comp. , 52 (2021), 177-185.
[8] Kátai I. and B. M. Phong, Arithmetical functions commutable with sums of squares, Notes on Number Theory and Discrete Mathematics, Volume 27, 2021, Number 3, 143-154.
[9] Kátai I. and B. M. Phong, Arithmetical functions commutable with sums of squares II, Mathematica Pannonica New Series, 27 /NS 1/ (2021) 2, 179-190.
[10] Khanh B. M. M. , On the equation f(n² + Dm²) = f(n)² + Df(m)², Ann. Univ. Sci. Budapest. Sect. Comput., 44 (2015), 59-68.
[11] Khanh B. M. M. , On conjecture concerning the functional equation, Annales Univ. Sci. Budapest., Sect. Comp. 46 (2017) 123-135.
[12] Khanh B. M. M. , A note on a result of B. Bojan, Annales Univ. Sci. Budapest., Sect. Comp. 49 (2019), 285-297 .
[13] Khanh B. M. M. , On the equation f(n²+Dm²+k) = f(n)²+Df(m)²+ k, Annales Univ. Sci. Budapest., Sect. Comp. 52 (2021),217-241.
[14] Poo-Sung Park, Multiplicative function commutable with sums of squares, International Journal of Number Theory, Vol. 14, No. 02 (2018), 469-478.
[15] Poo-Sung Park, On k-additive uniqueness of the set of squares for multiplicative functions , Aequationes mathematicae volume 92 (2018), 487-495.
[2] Chung P. V, Multiplicative functions satisfying the equation f(n² + m²) = f(n²) + f(m²), Mathematica Slovaca, 46 (1996), 165-171.
[3] Fehér J. and I. K atai, Sets of uniqueness for additive and multiplicative functions, Ann. Univ. Sci. Budapest. Sect. Math., 47 (2002), 3-16.
[4] Fehér J., K.-H. Indlekofer and N. M. Timofeev, A set of uniqueness for completely additive arithmetic functions, Annales Univ. Sci. Budapest. Sect. Comp., 21(2004), 57-67.
[5] I. Kátai and B. M. Phong, The functional equation f(A+B) = g(A)+ h(B), Annales Univ. Sci. Budapest. Sect. Comp., 43 (2014), 287-301.
[6] Kátai I. and B. M. Phong, Some unsolved problems on arithmetical functions, Ann. Univ. Sci. Budapest. Sect. Comput., 44 (2015), 233-235.
[7] Kátai I. and B. M. Phong, A characterization of functions using Lagrange's Four-Square Theorem, Annales Univ. Sci. Budapest., Sect. Comp. , 52 (2021), 177-185.
[8] Kátai I. and B. M. Phong, Arithmetical functions commutable with sums of squares, Notes on Number Theory and Discrete Mathematics, Volume 27, 2021, Number 3, 143-154.
[9] Kátai I. and B. M. Phong, Arithmetical functions commutable with sums of squares II, Mathematica Pannonica New Series, 27 /NS 1/ (2021) 2, 179-190.
[10] Khanh B. M. M. , On the equation f(n² + Dm²) = f(n)² + Df(m)², Ann. Univ. Sci. Budapest. Sect. Comput., 44 (2015), 59-68.
[11] Khanh B. M. M. , On conjecture concerning the functional equation, Annales Univ. Sci. Budapest., Sect. Comp. 46 (2017) 123-135.
[12] Khanh B. M. M. , A note on a result of B. Bojan, Annales Univ. Sci. Budapest., Sect. Comp. 49 (2019), 285-297 .
[13] Khanh B. M. M. , On the equation f(n²+Dm²+k) = f(n)²+Df(m)²+ k, Annales Univ. Sci. Budapest., Sect. Comp. 52 (2021),217-241.
[14] Poo-Sung Park, Multiplicative function commutable with sums of squares, International Journal of Number Theory, Vol. 14, No. 02 (2018), 469-478.
[15] Poo-Sung Park, On k-additive uniqueness of the set of squares for multiplicative functions , Aequationes mathematicae volume 92 (2018), 487-495.