What function can produce a perfect saddleback plot and fulfil the following requirement? I need to find a function that produce a good saddleback plot. The function has the following requirements: 1. Having 2 arguments: x and y 2. Both x and y are natural numbers 3. The result of the function is natural number 4. The function is increasing in each argument In order to plot a saddleback (in 3D), I tried $ x^2 - y^2 $, which gives such plot: !enter image description here It looks like a saddleback, but **it doesn't fit to the requirements**. I then tried $ 3x+27y+y^2 $, it gives plot like this: !enter image description here It is not that _saddleback_ , right? So can anyone supply me a good one on this?

What function can produce a perfect saddle shape?

I propose function $f(x,y)=x y $. It fulfills all the arithmetic conditions, and its plot has the shape of a saddle (‘hyperbolic paraboloid’) because, as a quadratic form it has signature (1,1), i.e. is the sum of a definite positive and a definite negative form:

$$ xy=\frac14(x+y)^2-\frac14(x-y)^2. $$

So it may be the same or some rotated shape.

Please suggest any equations to other such saddle shapes.

!Hypar 3D graph