derivation of Hartley function
Let be a function on positive integers that satisfies the above three properties. Using the additive property, it is easy to see that the value of must be zero. So we want to show that for all integers .
From the additive property, we can show that for any integer and ,
Let be an integer. Let be any positive integer. There is a unique integer determined by
On the other hand, by monotonicity,
Using Equation (1) and , we get
Since can be arbitrarily large, the difference on the left hand of the above inequality must be zero,