Arithmetic Differential Equations Defined by Generalized Subderivatives

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.