# Arithmetic derivative

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
View a complete list of arithmetic functions

## Definition

The arithmetic derivative or number derivative is an arithmetic function, specifically a function from  to  denoted by the  superscript, defined in a number of equivalent ways.

Definition type Definition details
using Leibniz rule and specification on primes It is defined by the following three conditions:

 for any prime number 
Leibniz rule:  for any (possibly equal, possibly distinct) natural numbers 
direct definition in terms of prime factorization Consider a natural number  with prime factorization  where the  are all distinct primes and the  are all positive integers (possibly repeated). Then the arithmetic derivative  is given by 

## Higher derivatives

Note that for any , the arithmetic derivative of  is nonzero, so the arithmetic derivative operation can be iterated for . We can thus consider iterations of the arithmetic derivative operation, which are denoted by using multiple primes. Note that higher derivatives make sense only as long as we do not hit zero.

The second derivative of , for instance, is denoted .

## Relation with conjectures

Conjecture What the conjecture, if true, would imply about the arithmetic derivative Explanation Is the converse implication true?
Goldbach's conjecture: every even integer greater than 2 is expressible as a sum of two primes. For every even integer  with , there exists an integer  such that . If , then we can take . Note that this construction works regardless of whether  are equal or distinct. Not obviously so
twin primes conjecture: there exist infinitely many primes  for which  is prime. There exist infinitely many integers  such that . For any prime  such that  is prime,  satisfies . Not obviously so