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# Arithmetic derivative

In number theory, the **arithmetic derivative**, or number derivative, is an arithmetic function defined for natural numbers, based on their prime factorization, and a product rule by analogy with the product rule for the derivative of a function that is used in analysis.

## Contents

- 1 Arithmetic derivative of natural numbers
- 2 Arithmetic logarithmic derivative
- 3 Arithmetic derivative of rational numbers
- 4 Sequences
- 5 Arithmetic derivative of Gaussian integers
- 6 Arithmetic derivative of Gaussian rationals
- 7 Generalizations
- 8 Applications
- 9 See also
- 10 References
- 11 External links

## Arithmetic derivative of natural numbers

### Definition

- For prime numbers, the arithmetic derivative is defined as

for any prime

- For natural numbers, the arithmetic derivative is defined in terms of their prime factorization, using the product rule

for any .

Considering a natural number's prime factorization

where are the distinct prime factors of , ω(n) is the number of distinct prime factors of and are positive integers,

its arithmetic derivative is thus given by

### Arithmetic derivative of 1

By the Leibniz rule

thus

Also

since the empty sum is 0.

### Arithmetic derivative of -1

By the Leibniz rule

thus

### Arithmetic derivative of -n

By the Leibniz rule

thus

### Arithmetic derivative of 0

Unlike the functional derivative of analysis, the arithmetic derivative of a sum is not the sum of the arithmetic derivatives, so the following

thus

is a fallacious 'proof'. But the result is true:

thus

so

E. J. Barbeau was the first to formalize this definition. He extended it to all integers by proving that uniquely defines the derivative over the integers. Barbeau also further extended it to rational numbers. Victor Ufnarovski and Bo Åhlander expanded it to certain algebraic numbers. In these extensions, the formula above still applies, but the exponents are allowed to be arbitrary rational numbers.

### Properties

#### Arithmetic derivative of 0 or a unit (1, -1)

The arithmetic derivative of is zero if and only if is zero or a unit (i.e., an invertible element).

#### Arithmetic derivative of integer primes

An integer prime is a unit (1, -1) times a positive prime .

The arithmetic derivative of an integer prime (a unit (1, -1) times a positive prime ) is that unit (if and only if rule)

#### Arithmetic second derivative of integer primes

The arithmetic second derivative of an integer prime is thus zero (if and only if rule)

#### Power rule (for prime powers)

The arithmetic derivative preserves the power rule (for prime powers):

where is prime and is a positive integer.

## Arithmetic logarithmic derivative

so we may define the arithmetic logarithmic derivative of as

## Arithmetic derivative of rational numbers

For any nonzero

thus

For any (quotient rule), we have

- ,

which is analogous to the quotient rule for the derivative of a function in analysis.

### Properties

The arithmetic derivative preserves the power rule (for negative prime powers):

where is prime and

## Sequences

The arithmetic derivative of , gives the sequence A003415

- {0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 10, 13, 1, 44, 10, 15, 27, 32, 1, 31, 1, 80, 14, 19, 12, 60, 1, 21, 16, 68, 1, 41, 1, 48, 39, 25, 1, 112, 14, 45, 20, 56, 1, 81, 16, 92, 22, 31, 1, 92, 1, 33, 51, ...}

## Arithmetic derivative of Gaussian integers

Ufnarovski and Ahlander briefly mention this idea, but they do not pursue it because the derivative of Gaussian integers is not an extension of the arithmetic derivative of integers. Every nonzero Gaussian integer has a unique factorization into the product of a unit (1, -1, i, -i) and powers of positive Gaussian primes (i.e. Gaussian primes where and ).

### Definition

- The arithmetic derivative of all positive (first quadrant) Gaussian primes is defined as 1

- The arithmetic derivative of a product follows the Leibniz rule

### Arithmetic derivative of 0 or a unit (1, -1, i, -i)

The definition leads to the following results

- The arithmetic derivative of 0 or a unit (1, -1, i, -i) is 0;

- The arithmetic derivative of a unit (1, -1, i, -i) times a Gaussian integer is that unit times the arithmetic derivative of that Gaussian integer

### Arithmetic derivative of Gaussian primes

A Gaussian prime is a unit (1, -1, i, -i) times a positive Gaussian prime.

The arithmetic derivative of a Gaussian prime (a unit (1, -1, i, -i) times a positive Gaussian prime) is that unit (if and only if rule)

### Arithmetic second derivative of Gaussian primes

The arithmetic second derivative of a Gaussian prime is thus zero (if and only if rule)

## Arithmetic derivative of Gaussian rationals

This definition of arithmetic derivative can be extended to fractions , where and are Gaussian integers.

## Generalizations

The relation implies , but it does not imply for a prime. In fact, any function defined on the primes can be extended uniquely to a function on the integers satisfying this relation (Cf. A003415, comment from: Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 07 2006):

## Applications

Emmons, Krebs, & Shaheen suggest several undergraduate projects involving the study of the arithmetic derivative.^{[1]}

## See also

- A003415 a(n) = n' = derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = m*a(n) + n*a(m).
- A086134 Least prime factor of n'.
- A086131 Greatest prime factor of n'.
- A085731 gcd(n, n').

- k-th arithmetic derivative of n
- A003415 a(n) = n' = derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = m*a(n) + n*a(m).
- A068346 Arithmetic second derivative of n.
- A099306 Arithmetic third derivative of n.
- ...
- A129150 n-th arithmetic derivative of 2^3.
- A129151 n-th arithmetic derivative of 3^4.
- A129152 n-th arithmetic derivative of 5^6.
- A099309 k-th arithmetic derivative of n is nonzero for all k.
- A099308 k-th arithmetic derivative of n is zero for some k.
- A099307 Least k such that the k-th arithmetic derivative of n is zero.

- Arithmetic derivative of polynomials

- A068329 Arithmetic derivative of Fibonacci(n).

- Arithmetic derivative of exponentials

- Arithmetic derivative of super-exponential functions

- Arithmetic derivative of special numbers

- Arithmetic derivative of arithmetic functions

- 'Arithmetic differential' Diophantine equations
- A051674 n such that n' = n (i.e. (n-th prime)^(n-th prime).)
- A098700 n such that x' = n has no integer solution.
- A098699 Least x such that x' = n.
- A099303 Greatest x such that x' = n.
- A099302 Number of solutions to x' = n.
- A099304 Least such that (n+k)' = n' + k'.
- A099305 Number of solutions to (n+k)' = n' + k'.

- Arithmetic derivative of rational numbers

- Arithmetic derivative of Gaussian integers

## References

- E. J. Barbeau, "Remark on an arithmetic derivative",
*Canadian Mathematical Bulletin,*Vol. 4 (1961), 117–122. - Victor Ufnarovski and Bo Åhlander, "How to Differentiate a Number",
*Journal of Integer Sequences*Vol. 6 (2003), Article 03.3.4.

- ↑ Caleb Emmons, Mike Krebs, Anthony Shaheen, How to differentiate an integer modulo n,
*The College Mathematics Journal***40**:5, pp. 345-353 (2009).

## External links

- Arithmetic Derivative
*, Planet Math*, accessed 04:15, 9 April 2008 (UTC). - L. Westrick,
*Investigations of the Number Derivative*. - I. Peterson,
*Math Trek: Deriving the Structure of Numbers*. - Michael Stay,
*Generalized Number Derivatives,*Journal of Integer Sequences, Vol. 8 (2005).