I was testing some data on the positive integer representations of $f(x, y) = x^3 - y^2$ such that $(x, y) \in [1, M] \times [1, M]$. I tested it for $M = 1000, 10000$. It turns out that the density function $P(f(x, y) = n) \sim n^{-0.7}$.
I wanted to know if there are any know results in this direction. I know that there are some results of Hooley on the representation of the sum of $h$th powers. But my particular question seems to be significantly different.
