Arithmetic Differential Equations Defined by 
Generalized Subderivatives

Champak Talukdar; Helen K. Saikia

doi:https://doi.org/10.14445/22315373/IJMTT-V72I1P104

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 1 | Year 2026 | Article Id. IJMTT-V72I1P104 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I1P104

Arithmetic Differential Equations Defined by Generalized Subderivatives

Champak Talukdar, Helen K. Saikia

Received	Revised	Accepted	Published
18 Nov 2025	27 Dec 2025	13 Jan 2026	28 Jan 2026

Citation :

Champak Talukdar, Helen K. Saikia, "Arithmetic Differential Equations Defined by Generalized Subderivatives," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 1, pp. 21-27, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I1P104

Abstract

The arithmetic derivative, introduced by Barbeau (1961), is an integer-valued analogue of the usual derivative. For a nonempty set 𝑆 of primes, the arithmetic subderivative 𝐷𝑆 differentiates an integer only with respect to the primes in 𝑆, treating primes outside 𝑆 as constants. The present work studies arithmetic differential equations on the integers defined by 𝐷𝑆 and its iterates. Criteria are obtained for the vanishing of iterates 𝐷𝑆𝑘(𝑛), in terms of the S-support of intermediate values; in particular, D𝑆2(𝑛) = 0 occurs exactly when 𝐷𝑆(𝑛) is free of primes from S. Fixed points of 𝐷𝑆 are also determined: a positive integer n satisfies 𝐷𝑆(𝑛) = 𝑛 precisely when 𝛴𝑝∈𝑆𝜈𝑝(𝑛)/𝑝 = 1, which forces 𝑛 = 𝑝𝑝𝑚, where 𝑝 ∈ 𝑆 and 𝑚 has no prime factor in 𝑆. Examples are included to illustrate both terminating and non-terminating trajectories for different choices of 𝑆.

Keywords

Arithmetic derivative, Arithmetic subderivatives, Fixed points, Leibniz rule, Prime factorization.

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