The Cantor pairing function is a primitive recursive pairing function defined by The statement that this is the only quadratic pairing function is known as the Fueter–Pólya theorem. Whether this is the only polynomial pairing function is still an open question. When we apply the pairing function to and we often denote the resulting number as. This definition can be inductively generalized to the Cantor tuple function for as with the base case defined above for a pair:
Inverting the Cantor pairing function
Let be an arbitrary natural number. We will show that there exist unique values such that and hence that is invertible. It is helpful to define some intermediate values in the calculation: where is the triangle number of. If we solve the quadratic equation for as a function of, we get which is a strictly increasing and continuous function when is non-negative real. Since we get that and thus where is the floor function. So to calculate and from, we do: Since the Cantor pairing function is invertible, it must be one-to-one and onto.
Examples
To calculate : so. To find and such that : so ; so ; so ; so ; thus.
Derivation
The graphical shape of Cantor's pairing function, a diagonal progression, is a standard trick in working with infinite sequences and countability. The algebraic rules of this diagonal-shaped function can verify its validity for a range of polynomials, of which a quadratic will turn out to be the simplest, using the method of induction. Indeed, this same technique can also be followed to try and derive any number of other functions for any variety of schemes for enumerating the plane. A pairing function can usually be defined inductively – that is, given the th pair, what is the th pair? The way Cantor's function progresses diagonally across the plane can be expressed as The function must also define what to do when it hits the boundaries of the 1st quadrant – Cantor's pairing function resets back to the x-axis to resume its diagonal progression one step further out, or algebraically: Also we need to define the starting point, what will be the initial step in our induction method:. Assume that there is a quadratic 2-dimensional polynomial that can fit these conditions. The general form is then Plug in our initial and boundary conditions to get and: so we can match our terms to get So every parameter can be written in terms of except for, and we have a final equation, our diagonal step, that will relate them: Expand and match terms again to get fixed values for and, and thus all parameters: Therefore is the Cantor pairing function, and we also demonstrated through the derivation that this satisfies all the conditions of induction.
