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| In the [[theory of computation]], the '''Sudan function''' is an example of a [[function (mathematics)|function]] that is [[recursion#Functional recursion|recursive]], but not [[primitive recursive function|primitive recursive]]. This is also true of the better-known [[Ackermann function]]. The Sudan function was the first function having this property to be published.
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| It was discovered (and published<ref>Bull. Math. Soc. Roumaine Sci. 30 (1927), 11 - 30; Jbuch 53, 171</ref>) in 1927 by [[Gabriel Sudan]], a [[Romania]]n [[mathematician]] who was a student of [[David Hilbert]].
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| ==Definition==
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| :<math>F _0 (x, y) = x+y,\,</math> | |
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| :<math>F _{n+1} (x, 0) = x, \ n \ge 0\,</math> | |
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| :<math>F _{n+1} (x, y+1) = F _n (F_{n+1} (x, y), F_{n+1} (x, y) + y + 1), \ n\ge 0.\,</math> | |
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| ==Value Tables==
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| {| class="wikitable"
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| |+ Values of ''F''<sub>1</sub>(''x'', ''y'')
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| |-
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| ! ''y''\''x''
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| ! 0
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| ! 1
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| ! 2
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| ! 3
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| ! 4
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| ! 5
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| |-
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| ! 0
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| | 0 || 1 || 2 || 3 || 4 || 5
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| |-
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| ! 1
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| | 1 || 3 || 5 || 7 || 9 || 11
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| |-
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| ! 2
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| | 4 || 8 || 12 || 16 || 20 || 24
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| |-
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| ! 3
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| | 11 || 19 || 27 || 35 || 43 || 51
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| |-
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| ! 4
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| | 26 || 42 || 58 || 74 || 90 || 106
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| |-
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| ! 5
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| | 57 || 89 || 121 || 153 || 185 || 217
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| |-
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| ! 6
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| | 120 || 184 || 248 || 312 || 376 || 440
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| |}
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| In general, ''F''<sub>1</sub>(''x'', ''y'') is equal to ''F''<sub>1</sub>(0, ''y'') + 2<sup>''y''</sup> ''x''.
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| {| class="wikitable"
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| |+ Values of ''F''<sub>2</sub>(''x'', ''y'')
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| |-
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| ! ''y''\''x''
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| ! 0
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| ! 1
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| ! 2
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| ! 3
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| ! 4
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| ! 5
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| |-
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| ! 0
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| | 0 || 1 || 2 || 3 || 4 || 5
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| |-
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| ! 1
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| | 1 || 8 || 27 || 74 || 185 || 440
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| |-
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| ! 2
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| | 19 || F<sub>1</sub>(8, 10) = 10228 || F<sub>1</sub>(27, 29) ≈ 1.55 {{e|10}}
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| | F<sub>1</sub>(74, 76) ≈ 5.74 {{e|24}}
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| | F<sub>1</sub>(185, 187) ≈ 3.67 {{e|58}}
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| | F<sub>1</sub>(440, 442) ≈ 5.02 {{e|135}}
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| |}
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| ==References==
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| *Cristian Calude, [[Solomon Marcus]], Ionel Tevy, ''The first example of a recursive function which is not primitive recursive'', Historia Mathematica 6 (1979), no. 4, 380–384 {{doi|10.1016/0315-0860(79)90024-7}}
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| <references/>
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| {{DEFAULTSORT:Sudan Function}}
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| [[Category:Arithmetic]]
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| [[Category:Large integers]]
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| [[Category:Special functions]]
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| [[Category:Theory of computation]]
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| {{mathlogic-stub}}
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