I am no math expert. I do not understand Mandelbrot set and its properties and applications. I know this puzzle is not difficult once you got some basic math fundamentals.
So the hardest (sometimes boring) parts come first:
First, you should study the definition of Mandelbrot set. Got an overview of what it is. (google, wiki, many edu and math web sites)
Second, by step 1, you should know that Mandelbrot set involves Complex number operations. If you do not know what is Complex number, google again to understand it.
Third, by step 1 and 2 and by the puzzle statement, you should have an equation about Mandelbrot set. Google again if you need more understanding of the equation.
Fourth, by the above studies, you should plan how to implement Complex number calculations in your programming language. For example in Java, I created a ComplexNumber class, under which I have constructors for creating Complex number objects. I have square and add methods for calculation.
I have a method to test whether the calculation will result in “diverge” status. (Don’t know what is diverge? Google helps.)
Putting them together is easy.