pairing function

we define the primitive recursive function
$\text{[math]}$ if $\text{[math]}$ is any given number, there is a unique solution $\text{[math]}$ to the equation
$\text{[math]}$ namely, $\text{[math]}$ is the largest number such that $\text{[math]}$ , and $\text{[math]}$ is then the solution of the equation
$\text{[math]}$ this last equation has a (unique) solution because $\text{[math]}$ must be odd. (the twos have been "divided out.") pairing_function.html thus defines functions
$\text{[math]}$ since pairing_function.html implies that $\text{[math]}$ we have
$\text{[math]}$ hence we can write
$\text{[math]}$ so that $\text{[math]}$ are primitive recursive functions.
the definition of $\text{[math]}$ can be expressed by the statement
$\text{[math]}$ the properties of the functions $\text{[math]}$ , and $\text{[math]}$ are summarized in pairing_function.html.
[cite:;taken from @computability_davis_1994 chapter 8 pairing functions and godel numbers]

the functions $\text{[math]}$ , and $\text{[math]}$ have the following properties:

they are primitive recursive;
$\text{[math]}$ ;
$\text{[math]}$ ;
$\text{[math]}$ .

[cite:;taken from @computability_davis_1994 chapter 3 primitive recursive functions; theorem 8.1]