I find these incredibly interesting, maybe lots of number theory problems could be solved by solving arithmetic differential equations.

They're defined like this: ($D(x)$ denotes the arithmetic derivative of an $x \in \N$)

If x is prime

$$D(x) = 1[/math]<br /> <br /> Else:<br /> <br /> <script type="math/tex; mode=display">D(xy) = D(x)y + xD(y)$$

Even things like the power rule hold up:

$$D(x^n) = nx^{n-1}D(x)$$

Some major number theory problems can be easily expressed using this, such as the goldbach conjecture, that in terms of arithmetic derivatives states that:

$$\forall k &gt; 1 \exists n : D(n) = 2k$$

The twin prime conjecture can be expressed as an arithmetic Differential equation as well, by stating that there are infinitely many $k$ such that:

$$D^2(k) = 1$$

Thought this was interesting, thoughts?