Arithmetic derivative

In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis. There are many versions of "arithmetic derivatives", including the one discussed in this article (the Lagarias arithmetic derivative), such as Ihara's arithmetic derivative and Buium's arithmetic derivatives. The arithmetic derivative was introduced by Spanish mathematician Josè Mingot Shelly in 1911. The arithmetic derivative also appeared in the 1950 Putnam Competition. For natural numbers n, the arithmetic derivative D(n) is defined as follows: D(p) = 1 for any prime p. D(mn) = D(m)n + mD(n) for any (Leibniz rule). Edward J. Barbeau extended the domain to all integers by showing that the choice D(−n) = −D(n), which uniquely extends the domain to the integers and is consistent with the product formula. Barbeau also further extended it to the rational numbers, showing that the familiar quotient rule gives a well-defined derivative on : Victor Ufnarovski and Bo Åhlander expanded it to the irrationals that can be written as the product of primes raised to arbitrary rational powers, allowing expressions like to be computed. The arithmetic derivative can also be extended to any unique factorization domain (UFD), such as the Gaussian integers and the Eisenstein integers, and its associated field of fractions. If the UFD is a polynomial ring, then the arithmetic derivative is the same as the derivation over said polynomial ring. For example, the regular derivative is the arithmetic derivative for the rings of univariate real and complex polynomial and rational functions, which can be proven using the fundamental theorem of algebra. The arithmetic derivative has also been extended to the ring of integers modulo n. The Leibniz rule implies that D(0) = 0 (take m = n = 0) and D(1) = 0 (take m = n = 1). The power rule is also valid for the arithmetic derivative.