 On this basis, we may now define a variant of the so-called Ackermann-Péter function as \(\pi(i,x) = \beta(i)(x)\)—i.e., the result of applying the function \(\beta(i)\) to the argument \(x\). \(\pi(i,x)\) has the same order of growth as \(\alpha_i(x,x)\) and it is possible to prove via the argument sketched above that \(\pi(i,x)\) is not primitive recursive (see, e.g., Péter 1967, ch. 9). Based on earlier work of Ackermann (1928), Péter (1935) also showed that \(\pi(i,x)\) may also be defined by a so-called doubly recursive definition of the following form which takes only natural numbers as  arguments:[9] 