# arithmetic derivative

## Primary tabs

Type of Math Object:
Definition
Major Section:
Reference

## Mathematics Subject Classification

11Z05 Miscellaneous applications of number theory

### What is it good for?

While browsing the encyclopaedia, I came across the definition of derivative of a number. It is certainly an inteteresting illustration of how Leibniz's rule can be used to define a notion of derivative in a context where one would ordinarily not think of differential calculus. My question to you is "What is this notion of integer derivative good for?", meaning "Are you aware of any theorems of number theory which can be proven using the integer derivative?".

### Re: What is it good for?

... when the derivative is zero, the number is a
constant :-)

Matte

### Re: What is it good for?

What exactly do you mean by this??? I would like to think that every single integer is a constant.

Johan

### Re: What is it good for?

Of course every integer is a constant.
My comment was just a joke.
Nevertheless, I'm too curious how
this integer derivative is used.

Matte

### Re: What is it good for?

Apparently Goldbach's conjecture can be stated using
the number derivative as:

if a\in Z then there exists a b\in Z such that
b'=2a.

The number derivative is also related to the
twin prime conjecture:

Matte

### Re: What is it good for?

It's unfortunate that this number derivative is not compatible with addition. In general (n+m)' != n' + m'. If it were, it would have made a nice example of a differential ring (field) if extended to the integers (rationals).