Arithmetic derivative
From Number
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 |
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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 |