Some questions related to the arithmetic derivative and its applications
https://doi.org/10.22405/2226-8383-2025-26-3-96-112
Abstract
The article discusses the properties of the arithmetic derivative and the primitive on the set of rational numbers, obtained properties related to similar properties of the derivative function, some equations in arithmetic derivatives and the properties of their solutions.
About the Authors
Elena Ivanovna Deza
Institute of Mathematics and Computer Science, Moscow Pedagogical State University
Russian Federation
Russian Federation
doctor of pedagogical sciences, professor
Tatiana Nikolaevna Kazarikhina
Institute of Mathematics and Computer Science, Moscow Pedagogical State University
Russian Federation
Russian Federation
candidate of pedagogical sciences
Darya Yurievna Ganusenko
Institute of Mathematics and Computer Science, Moscow Pedagogical State University
Russian Federation
Russian Federation
Sergey Olegovich Tikhonov
Institute of Mathematics and Computer Science, Moscow Pedagogical State University
Russian Federation
Russian Federation
References
1. Bukhshtab A.A. 2024, Number Theory. Saint Petersburg: Lan. 384 p.
2. Petrovsky I.G. 2022, Lectures on the Theory of Ordinary Differential Equations. Moscow: URSS. 240 p.
3. Pontryagin L.S. 2023, Ordinary Differential Equations. Moscow: URSS. 336 p.
4. Barbeau E.J. 1961, “Remarks on arithmetic derivative”, Canadian Mathematical Bulletin 4(2) pp. 117–122.
5. Buium A. 1977, “Arithmetic Analogues of Derivations”, Journal of Algebra 198 pp. 290–299.
6. Deza E.I. 2023, Perfect and Amicable Numbers. Singapore: World Scientific. 462 p.
7. Dahl N., Olsson J., Loiko A. 2011, “Investigations on the properties of the arithmetic derivative”, arXiv [Preprint]. Available at: https://arxiv.org/pdf/1108.4762 (Accessed: 02.07.2025)
8. Haukkanen P., Merikoski J.K., Tossavainen T. 2018, “The arithmetic derivative and Leibnizadditive functions”, arXiv [Preprint]. Available at: https://arxiv.org/pdf/1803.06849 (Accessed: 02.07.2025)
9. Koviˇc J. 2015, “The Arithmetic Derivative and Antiderivative”, Journal of Integer Sequences [Online] 16. Available at: https://cs.uwaterloo.ca/journals/JIS/VOL15/Kovic/kovic4.html (Accessed: 02.07.2025)
10. Merikoski J.K., Haukkanen P., Tossavainen T. 2021, “Complete additivity, complete multiplicativity, and Leibniz-additivity on rationals”, Integers [Online] 21. Available at: https://math.colgate.edu/ integers/v33/v33.pdf (Accessed: 02.07.2025)
11. Stay M. 2005, “Generalized Number Derivatives”, Journal of Integer Sequences [Online] 8. Available at: https://cs.uwaterloo.ca/journals/JIS/VOL8/Stay/stay44.html (Accessed: 02.07.2025)
12. Tossavainen T., Haukkanen P., Merikoski J.K., Mattila M. 2024, “We can differentiate numbers, too”, The College Mathematics Journal 55(2) pp. 100–108.
13. Ufnarovski V., ˚Ahlander B. 2003, “How to differentiate a number”, Journal of Integer Sequences [Online] 6. Available at: https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html (Accessed: 02.07.2025)
14. Westrick L. 2003, “Investigations of the Number Derivative”, Internet Archive [Online]. Available at: https://web.archive.org/web/20050426071741/http://web.mit.edu/lwest/www/intmain.pdf (Accessed: 02.07.2025)
Review
For citations:
Deza E.I., Kazarikhina T.N., Ganusenko D.Yu., Tikhonov S.O. Some questions related to the arithmetic derivative and its applications. Chebyshevskii Sbornik. 2025;26(3):96-112. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-3-96-112
Views:
26
Email this article 