Sudan function

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[1]) in 1927 by Gabriel Sudan, a Romanian mathematician who was a student of David Hilbert.

Definition

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

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

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

Value tables

Values of F₀(x, y)

y\x	0	1	2	3	4	5
0	0	1	2	3	4	5
1	1	2	3	4	5	6
2	2	3	4	5	6	7
3	3	4	5	6	7	8
4	4	5	6	7	8	9
5	5	6	7	8	9	10
6	6	7	8	9	10	11

Values of F₁(x, y)

y\x	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
0	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
1	1	3	5	7	9	11	13	15	17	19	21	23	25	27	29
2	4	8	12	16	20	24	28	32	36	40	44	48	52	56	60
3	11	19	27	35	43	51	59	67	75	83	91	99	107	115	123
4	26	42	58	74	90	106	122	138	154	170	186	202	218	234	250
5	57	89	121	153	185	217	249	281	313	345	377	409	441	473	505
6	120	184	248	312	376	440	504	568	632	696	760	824	888	952	1016
7	247	375	503	631	759	887	1015	1143	1271	1399	1527	1655	1783	1911	2039
8	502	758	1014	1270	1526	1782	2038	2294	2550	2806	3062	3318	3574	3830	4086
9	1013	1525	2037	2549	3061	3573	4085	4597	5109	5621	6133	6645	7157	7669	8181
10	2036	3060	4084	5108	6132	7156	8180	9204	10228	11252	12276	13300	14324	15348	16372
11	4083	6131	8179	10227	12275	14323	16371	18419	20467	22515	24563	26611	28659	30707	32755
12	8178	12274	16370	20466	24562	28658	32754	36850	40946	45042	49138	53234	57330	61426	65522
13	16369	24561	32753	40945	49137	57329	65521	73713	81905	90097	98289	106481	114673	122865	131057
14	32752	49136	65520	81904	98288	114672	131056	147440	163824	180208	196592	212976	229360	245744	262128

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 ×10^10	F1(74, 76) ≈ 5.74 ×10^24	F1(185, 187) ≈ 3.67 ×10^58	F1(440, 442) ≈ 5.02 ×10^135

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 doi:10.1016/0315-0860(79)90024-7

1. Bull. Math. Soc. Roumaine Sci. 30 (1927), 11 - 30; Jbuch 53, 171

External links

• OEIS: A260003, A260004