The Arithmetic Partial Derivative
Brad Emmons and Xiao Xiao
Department of Mathematics
Utica, NY 13502
The arithmetic partial derivative (with respect to a prime p) is a
function from the set of integers that sends p to 1 and satisfies
the Leibniz rule. In this paper, we prove that the p-adic valuation
of the sequence of higher order partial derivatives is eventually
periodic. We also prove a criterion to determine when an integer has
integral anti-partial derivatives. As an application, we show that
there are infinitely many integers with exactly n integral anti-partial
derivatives for every nonnegative integer n.
Full version: pdf,
(Concerned with sequences
Received January 29 2022; revised version received June 1 2022.
Published in Journal of Integer Sequences,
June 4 2022.
Journal of Integer Sequences home page