Editing Ackermann function (section)

{{Short description|Quickly growing function}}
{{About|the mathematical function||Ackermann (disambiguation)}}
{{Use dmy dates|date=October 2024}}
{{Use shortened footnotes|date=November 2022}}
In [[computability theory]], the '''Ackermann function''', named after [[Wilhelm Ackermann]], is one of the simplest{{sfn|Monin|Hinchey|2003|p=61}} and earliest-discovered examples of a [[total function|total]] [[computable function]] that is not [[Primitive recursive function|primitive recursive]]. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive.

After Ackermann's publication{{sfn|Ackermann|1928}} of his function (which had three non-negative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann function" may refer to any of numerous variants of the original function. One common version is the two-argument '''Ackermann–Péter function''' developed by [[Rózsa Péter]] and [[Raphael Robinson]]. This function is defined from the [[recurrence relation]] <math>\operatorname{A}(m+1, n+1) = \operatorname{A}(m, \operatorname{A}(m+1, n))
</math> with appropriate [[Base case (recursion)|base cases]]. Its value grows very rapidly; for example, <math>\operatorname{A}(4, 2)</math> results in <math>2^{65536} - 3</math>, an integer with 19,729 decimal digits.<ref>{{cite web |title=Decimal expansion of A(4,2)|archive-url= https://web.archive.org/web/20100120134707/http://kosara.net/thoughts/ackermann42.html |date=August 27, 2000| url= http://www.kosara.net/thoughts/ackermann42.html|archive-date=January 20, 2010|website=kosara.net }}</ref>