Tables of Exceptions

Given a prime p ≥ 3 and a base b ≥ 2 that are relatively prime, the decimal expansion of 1/p in base b has a repeat length n that satisfies this equation.

bⁿ ≡ 1 mod p

If the following stronger condition holds for some d ≥ 1, I call p an exception of degree d.

bⁿ ≡ 1 mod p^d+1

The tables below include all the exceptions that I mentioned in The Usual Random Thoughts. I just added a few more bases and included the values of n, which are sometimes useful.

From an article by Helmut Richter, some exceptions with large degrees.

p	b	n	d
13	239	4	3
5	443	4	3
5	1068	4	5

From various sources, some exceptions with large primes.

p	b	n	d
1645333507	5	1645333506	1
1645333507	25	822666753	1
1645333507	125	548444502	1
46145917691	34	46145917690	1

Finally, the complete table of exceptions with p ≤ 10⁸ and b ≤ 128.

p	b	n	d
3	8	2	1
3	10	1	1
3	17	2	1
3	19	1	1
3	26	2	2
3	28	1	2
3	35	2	1
3	37	1	1
3	44	2	1
3	46	1	1
3	53	2	2
3	55	1	2
3	62	2	1
3	64	1	1
3	71	2	1
3	73	1	1
3	80	2	3
3	82	1	3
3	89	2	1
3	91	1	1
3	98	2	1
3	100	1	1
3	107	2	2
3	109	1	2
3	116	2	1
3	118	1	1
3	125	2	1
3	127	1	1
5	7	4	1
5	18	4	1
5	24	2	1
5	26	1	1
5	32	4	1
5	43	4	1
5	49	2	1
5	51	1	1
5	57	4	2
5	68	4	2
5	74	2	1
5	76	1	1
5	82	4	1
5	93	4	1
5	99	2	1
5	101	1	1
5	107	4	1
5	118	4	1
5	124	2	2
5	126	1	2
7	18	3	2
7	19	6	2
7	30	3	1
7	31	6	1
7	48	2	1
7	50	1	1
7	67	3	1
7	68	6	1
7	79	3	1
7	80	6	1
7	97	2	1
7	99	1	1
7	116	3	1
7	117	6	1
7	128	3	1
11	3	5	1
11	9	5	1
11	27	5	1
11	40	10	1
11	81	5	1
11	94	10	1
11	112	10	1
11	118	10	1
11	120	2	1
11	122	1	1
11	124	5	2
13	19	12	1
13	22	3	1
13	23	6	1
13	70	4	1
13	80	12	1
13	89	12	1
13	99	4	1
17	38	4	1
17	40	16	1
17	65	16	1
17	75	16	1
17	110	8	1
19	28	9	1
19	54	9	1
19	62	9	1
19	68	3	1
19	69	6	1
19	99	9	1
19	116	18	1
19	127	18	1
23	28	22	1
23	42	22	2
23	63	22	1
23	118	11	1
29	14	28	1
29	41	4	1
29	60	28	1
29	63	14	1
31	115	30	1
31	117	30	1
37	18	36	1
37	76	36	1
37	117	4	1
41	51	5	1
43	19	42	1
43	75	14	1
43	78	7	1
47	53	23	1
47	67	46	1
47	71	23	1
47	116	46	1
59	53	29	1
71	11	70	1
71	26	14	1
71	121	35	1
79	31	39	1
83	99	41	1
97	53	48	1
97	107	96	1
103	43	102	1
109	96	108	1
113	68	112	2
127	38	21	1
127	62	63	1
131	58	5	1
131	111	130	1
137	19	68	1
151	78	25	1
163	65	27	1
163	84	81	1
181	78	180	1
223	69	111	1
229	44	19	1
233	33	116	1
241	94	15	1
257	48	256	1
263	79	262	1
281	20	140	1
293	91	73	1
307	40	153	1
313	104	78	1
331	18	110	1
331	71	165	1
347	75	173	1
353	14	352	1
461	52	460	1
487	10	486	1
487	100	243	1
509	93	127	1
631	69	30	1
647	56	19	1
653	84	652	1
653	120	652	1
673	22	224	1
727	92	121	1
829	46	276	1
863	13	862	1
907	127	906	1
1093	2	364	1
1093	4	182	1
1093	8	364	1
1093	16	91	1
1093	32	364	1
1093	64	182	1
1093	128	52	1
1109	76	554	1
1163	78	581	1
1283	45	1282	1
1291	62	645	1
1613	35	403	1
1741	119	145	1
1949	54	974	1
1999	87	999	1
2137	95	2136	1
2693	12	2692	1
2741	68	137	1
2777	59	1388	1
2791	122	1395	1
3037	79	3036	1
3511	2	1755	1
3511	4	1755	1
3511	8	585	1
3511	16	1755	1
3511	32	351	1
3511	64	585	1
3511	128	1755	1
3571	35	1190	1
3761	108	3760	1
4871	83	487	1
5381	110	1345	1
5437	96	5436	1
5851	44	585	1
6343	80	906	1
6451	31	3225	1
7559	102	7558	1
7669	105	1917	1
8039	39	8038	1
8329	96	2082	1
9181	114	540	1
9221	93	9220	1
9241	76	4620	1
9431	110	9430	1
10271	108	5135	1
11779	85	5889	1
11813	102	2953	1
13691	83	6845	1
15061	95	15060	1
20101	84	134	1
20771	5	10385	1
20771	25	10385	1
20771	125	10385	1
22511	124	22510	1
25633	24	6408	1
28627	98	28626	1
29131	15	29130	1
30109	55	30108	1
31247	75	15623	1
32143	94	32142	1
32687	77	32686	1
33923	18	33922	1
34849	123	5808	1
36713	63	36712	1
40487	5	40486	1
40487	25	20243	1
40487	125	40486	1
46021	17	7670	1
46457	20	46456	1
47441	33	9488	1
47699	57	47698	1
48121	87	48120	1
48947	17	24473	1
56149	78	28074	1
66161	6	66160	1
66161	36	33080	1
66431	40	949	1
68239	86	68238	1
77867	37	77866	1
79399	106	79398	1
81551	93	40775	1
86197	57	28732	1
123653	12	123652	1
131759	45	1243	1
142963	70	142962	1
160541	30	8027	1
182111	117	182110	1
237977	104	118988	1
268573	67	134286	1
383951	92	15358	1
401771	63	40177	1
463033	94	231516	1
491531	7	245765	1
491531	49	245765	1
534851	6	106970	1
534851	36	53485	1
613181	107	122636	1
661049	76	17396	1
672799	106	336399	1
1006003	3	1006002	1
1006003	9	503001	1
1006003	27	335334	1
1006003	81	503001	1
1012573	79	1012572	1
1025273	41	1025272	1
1050139	101	1050138	1
1284043	18	428014	1
1595813	22	797906	1
1747591	13	873795	1
2074031	120	29629	1
2481757	23	827252	1
2503037	69	2503036	1
2806861	31	467810	1
2914393	97	971464	1
3152249	118	286568	1
3152573	6	788143	1
3152573	36	788143	1
7079771	56	7079770	1
7278001	55	1213000	1
9377747	20	9377746	1
10404887	118	10404886	1
12026117	92	12026116	1
12925267	96	6462633	1
13703077	23	13703076	1
13778951	127	6889475	1
18768727	92	1042707	1
20252173	109	10126086	1
24490789	103	24490788	1
42250279	58	7041713	1
53471161	5	13367790	1
53471161	25	6683895	1
53471161	125	4455930	1
56598313	10	56598312	1
56598313	100	28299156	1
60312841	79	634872	1
61001527	98	10166921	1
63061489	19	63061488	1
89351671	66	89351670	1