As first shown by Meyer and Ritchie, do-loops (which have a fixed iteration limit) are a special case of while-loops. A function that can be implemented using only do-loops is called primitive recursive. (In contrast, a computable function can be coded using a combination of for- and while-loops, or while-loops only.) Examples of primitive recursive functions include power, greatest common divisor, and p_n(the function giving the nth prime). The Ackermann function is the simplest example of a well-defined total function that is computable but not primitive recursive, providing a counterexample to the belief in the early 1900s that every computable function was also primitive recursive.

Ackermann function | computable function | general recursive function | recursive function | total function