Arithmetic Partial Derivative

Abstract: 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 any nonnegative integer n. (This is a joint work with Brad Emmons.)

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