Newton fractal

In the theory of computation, the Sudan function is an example of a function that is recursive, but not 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.

It was discovered (and published) in 1927 by Gabriel Sudan, a Romanian mathematician who was a student of David Hilbert.

Definition

$F_{0}(x,y)=x+y,\,$

$F_{n+1}(x,0)=x,\ n\geq 0\,$

$F_{n+1}(x,y+1)=F_{n}(F_{n+1}(x,y),F_{n+1}(x,y)+y+1),\ n\geq 0.\,$

Value Tables

Values of F₁(x, y)

y\x	0	1	2	3	4	5
0	0	1	2	3	4	5
1	1	3	5	7	9	11
2	4	8	12	16	20	24
3	11	19	27	35	43	51
4	26	42	58	74	90	106
5	57	89	121	153	185	217
6	120	184	248	312	376	440

In general, F₁(x, y) is equal to F₁(0, y) + 2^y x.

Values of F₂(x, y)

y\x	0	1	2	3	4	5
0	0	1	2	3	4	5
1	1	8	27	74	185	440
2	19	F1(8, 10) = 10228	F1(27, 29) ≈ 1.55 Template:E	F1(74, 76) ≈ 5.74 Template:E	F1(185, 187) ≈ 3.67 Template:E	F1(440, 442) ≈ 5.02 Template:E

References

• 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 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.

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