Développement décimal périodique de l’inverse d’un nombre premier

La période du développement décimal périodique d’un nombre rationnel est le cycle composé d’une séquence finie de chiffres qui se répète à l’infini.

L'inverse d'un nombre premier, noté 1/p possède un développement décimal périodique dont la longueur de la période est notée δ_(p)

Ceci exclut les nombres premiers 2 et 5 dont l'inverse ne possède pas de développement décimal périodique

Exemple :

1/7 = 0,142857 142857 142857...

δ₍₇₎ = 6

Classement des nombres premiers

Le tableau ci-dessous propose un classement des nombres premiers en fonction de la longueur de la période de leurs inverses.

Tableau des nombres premiers dont δ_p <101

1	3
2	11
3	37
4	101
5	41	271
6	7	13
7	239	4649
8	73	137
9	333667
10	9091
11	21649	513239
12	9901
13	53	79	265371653
14	909091
15	31	2906161
16	17	5882353
17	2071723	5363222357
18	19	52579
19	1111111111111111111
20	3541	27961
21	43	1933	10838689
22	23	4093	8779
23	11111111111111111111111
24	99990001
25	21401	25601	182521213001
26	859	1058313049
27	757	440334654777631
28	29	281	121499449
29	3191	16763	43037	62003	77843839397
30	211	241	2161
31	2791	6943319	57336415063790604359
32	353	449	641	1409	69857
33	67	1344628210313298373
34	103	4013	21993833369
35	71	123551	102598800232111471
36	999999000001
37	2028119	247629013	2212394296770203368013
38	909090909090909091
39	900900900900990990990991
40	1676321	5964848081
41	83	1231	538987	201763709900322803748657942361
42	127	2689	459691
43	173	1527791	1963506722254397	2140992015395526641
44	89	1052788969	1056689261
45	238681	4185502830133110721
46	47	139	2531	549797184491917
47	35121409	316362908763458525001406154038726382279
48	9999999900000001
49	505885997	1976730144598190963568023014679333
50	251	5051	78875943472201
51	613	210631	52986961	13168164561429877
52	521	265371653	1900381976777332243781
53	107	1659431	1325815267337711173	47198858799491425660200071
54	70541929	14175966169
55	1321	62921	83251631	1300635692678058358830121
56	7841	127522001020150503761
57	21319	10749631	3931123022305129377976519
58	59	154083204930662557781201849
59	2559647034361	4340876285657460212144534289928559826755746751
60	61	4188901	39526741
61	733	4637	329401	974293	1360682471	106007173861643	7061709990156159479
62	909090909090909090909090909091
63	10837	23311	45613	45121231	1921436048294281
64	19841	976193	6187457	834427406578561
65	162503518711	5538396997364024056286510640780600481
66	599144041	183411838171
67	493121	79863595778924342083	28213380943176667001263153660999177245677
68	28559389	1491383821	2324557465671829
69	277	203864078068831	1595352086329224644348978893
70	4147571	265212793249617641
71	241573142393627673576957439049	45994811347886846310221728895223034301839
72	3169	98641	3199044596370769
73	12171337159	1855193842151350117	49207341634646326934001739482502131487446637
74	7253	422650073734453	296557347313446299
75	151	4201	15763985553739191709164170940063151
76	722817036322379041	1369778187490592461
77	5237	42043	29920507	136614668576002329371496447555915740910181043
78	157	6397	216451	1058313049	388847808493
79	317	6163	10271	307627	49172195536083790769	3660574762725521461527140564875080461079917
80	5070721	19721061166646717498359681
81	163	9397	2462401	676421558270641	130654897808007778425046117
82	2670502781396266997	3404193829806058997303
83	3367147378267	9512538508624154373682136329	346895716385857804544741137394505425384477
84	226549	4458192223320340849
85	262533041	8119594779271	4222100119405530170179331190291488789678081
86	57009401	2182600451	7306116556571817748755241
87	4003	72559	310170251658029759045157793237339498342763245483
88	617	16205834846012967584927082656402106953
89	497867	103733951	104984505733	5078554966026315671444089	403513310222809053284932818475878953159
90	29611	3762091	8985695684401
91	547	14197	17837	4262077	43442141653	316877365766624209	110742186470530054291318013
92	1289	18371524594609	4181003300071669867932658901
93	900900900900900900900900900900990990990990990990990990990991
94	6299	4855067598095567	297262705009139006771611927
95	191	59281	63841	1289981231950849543985493631	965194617121640791456070347951751
96	97	206209	66554101249	75118313082913
97	12004721	846035731396919233767211537899097169	109399846855370537540339266842070119107662296580348039
98	197	5076141624365532994918781726395939035533
99	199	397	34849	362853724342990469324766235474268869786311886053883
100	60101	7019801	182521213001	14103673319201	78875943472201	1680588011350901

Factorisation de 1 + 10 + 10² + ... + 10ⁿ

Soit N₁=1, N₂=11, N₃=111, ..., N_k=$\sum_{i=0}^{k-1}10^i$ = 1 + 10 + 10² + 10³ + ... + 10^k-1 $pour k \in \N$

La factorisation de N_k peut se décomposer comme le produit de P x K x F^[1] ou

• P : produit des nombres premiers p dont δ_p=k
• K : produit des nombres premiers p dont δ_p est un diviseur de k
• F : produits des nombres p^q ou p est un nombre premier et q un exposant >0 $\in \N$ tel que :
  • p^q soit un diviseur de k
  • k ≥ δ _p x p

1. ↑ calculé pour k<101

Exemple pour k=6 :

Factorisation de 111111 (1+10+10²+10³+10⁴+10⁵)

• P = 7 x 13
• K = 11 x 37
• K = 3

111111 = 7 x 13 x 11 x 37 x 3

Exemple pour k = 78 :

Factorisation de 111111111111111111111111111111111111111111111111111111111111111111111111111111

• P = 157 x 6397 x 216451 x 1058313049 x 388847808493
• K = 11x37x7x13x53x79x265371653x859x1058313049x900900900900990990990991
• K = 3 x 13

Calcul des facteurs K et F

k	Diviseur de k	Equivalent en nombre premier p (δ(p) est un diviseur de k)	Coef. F
1
2
3			3
4	2	11
5
6	2-3	11x37	3
7
8	2-4	11x101
9	3	37	32
10	2-5	11x41x271
11
12	2-3-4-6	11x37x101x7x13	3
13
14	2-7	11x239x4649
15	3-5	37x41x271	3
16	2-4-8	11x101x73x137
17
18	2-3-6-9	11x37x7x13	32
19
20	2-4-5-10	11x101x41x271x9091
21	3-7	37x239x4649	3
22	2-11	11x21649x513239	11
23
24	2-3-4-6-8-12	11x37x101x7x13	3
25	5	41x271
26	2-13	11x53x79x265371653
27	3-9	37x333667	33
28	2-4-7-14	11x101x239x4649x909091
29
30	2-3-5-6-10-15	11x37x41x271x7x13x9091x31x2906161	3
31
32	2-4-8-16	11x101x73x137x17x5882353
33	3-11	37x21649x513239	3
34	2-17	11x2071723x5363222357
35	5-7	41x271x239x4649
36	2-3-4-6-9-12-18	11x37x101x7x13x333667x9901x19x52579	32
37
38	2-19	11x1111111111111111111
39	3-13	37x53x79x265371653	3
40	2-4-5-8-10-20	11x101x41x271x73x137x9091x3541x27961
41
42	2-3-6-7-14-21	11x37x7x13x239x4649x909091x43x1933x10838689	3X7
43
44	2-4-11-22	11x101x21649x513239x23x4093x8779	11
45	3-5-9-15	37x41x271x333667x31x2906161	32
46	2-23	11x11111111111111111111111
47
48	2-3-4-6-8-12-16-24	11x37x101x7x13x73x137x9901x17x5882353x99990001	3
49	7	239x4649
50	2-5-10-25	11x41x271x9091x21401x25601x182521213001
51	3-17	37x2071723x5363222357	3
52	2-4-13-26	11x101x53x79x265371653x859x1058313049
53
54	2-3-9-27	11x37x333667x757x440334654777631	33
55	5-11	41x271x21649x513239
56	2-4-7-8-14-28	11x101x239x4649x73x137x909091x29x281x121499449
57	3-19	37x1111111111111111111	3
58	2-29	11x3191x16763x43037x62003x77843839397
59
60	2-3-4-5-6-10-12-15-20-30	11x37x101x41x271x7x13x9091x9901x31x2906161x3541x27961x211x241x2161	3
61
62	2-31	11x2791x6943319x57336415063790604359
63	3-7-9-21	37x239x4649x333667x43x1933x10838689	32
64	2-4-8-16-32	11x101x73x137x17x5882353x353x449x641x1409x69857
65	5-13	41x271x9901
66	2-3-6-11-22-33	11x37x7x13x21649x513239x23x4093x8779x67x1344628210313298373	3X11
67
68	2-4-17-34	11x101x207123x5363222357x103x4013x21993833369
69	3-23	37x11111111111111111111111	3
70	2-5-7-10-14-35	11x41x271x239x4649x9091x909091x71x123551x102598800232111471
71
72	2-3-4-6-8-9-12-18-24-36	11x37x101x7x13x73x137x333667x9901x19x52579x99990001x999999000001	32
73
74	2-37	1x2028119x247629013x2212394296770203368013
75	3-5-15-25	37X41X271X31X2906161X99990001	3
76	2-4-19-38	11x101x1111111111111111111x909090909090909091
77	7-11	239x4649x21649x513239
78	2-3-6-13-26-39	11x37x7x13x53x79x265371653x859x1058313049x900900900900990990990991	3X13
79
80	2-4-5-8-10-16-20-40	11X101X41X271X73X137X9091X17X5882353X3541X27961X1676321X5964848081
81	3-9-27	37x333667x757x440334654777631	34
82	2-41	11x83x1231x538987x201763709900322803748657942361
83
84	2-3-4-6-7-12-14-21-28-42	11x37x101x7x13x239x4649x9901x909091x43x1933x10838689x29x281x121499449x127x2689x459691	3*7
85	5-17	41x271x2071723x5363222357
86	2-43	11x173x1527791x1963506722254397x2140992015395526641
87	3-29	37x3191x16763x43037x62003x77843839397	3
88	2-4-8-11-22-44	11x101x73x137x21649x513239x23x4093x8779x89x1052788969x1056689261	11
89
90	2-3-5-6-9-10-15-18-30-45	11x37x41x271x7x13x333667x9091x31x2906161x19x52579x211x241x2161x238681x4185502830133110721	3
91	7-13	239x4649x53x79x265371653
92	2-4-23-46	11x101x11111111111111111111111x47x139x2531x549797184491917
93	3-31	37x2791x6943319x57336415063790604359	3
94	2-47	11x35121409x316362908763458525001406154038726382279
95	5-19	41x271x1111111111111111111
96	2-3-4-6-8-12-16-24-32-48	11x37x101x7x13x73x137x9901x17x5882353x99990001x353x449x641x1409x69857x9999999900000001	3
97
98	2-7-14-49	11x239x4649x909091x505885997x1976730144598190963568023014679333
99	3-9-11-33	37x333667x21649x513239x67x1344628210313298373	3
100	2-4-5-10-20-25-50	11x101x41x271x9091x3541x27961x21401x25601x182521213001