The definition is as follows:
The Ackermann function grows extremely fast; A(4,2) has already 19,729 digits. This extreme growth can be exploited to show that the computable function f(n) = A(n, n) is not primitive recursive.
Intuitively, the Ackermann function defines generalizations of multiplication by two (iterated additions) and exponentiation with base 2 (iterated multiplications) to iterated exponentiation, iteration of this operation and so on. It can be most concisely expressed using [Knuth's up-arrow notation]?:
In 1928, Ackermann? considered a function A(m, n, p) of three variables, the p-fold iterated exponentiation of m with n, and proved that it is a recursive function which is not primitive recursive. This definition was later simplified by Rosza Peter and Raphael Robinson to the two-variable definition given above.