Computation for Proposition 2

c1-c2[-(CPU time)] and results table for 5 ≤ e ≤ 12,

where
c1 = |Ae,s|, c2 = |Ae,s∩B|,
Ae,s = { (L,p) | 2s-1<L<2s, L is a prime number, p = 2e+1L+1 is a prime number },
B = { (L,p) | 2 is a primitive root modulo L }.

In the range, L≤228 and p≤250, the total of c1 is 12,526,998 and the total of c2 is 4,618,127. In all cases, ¬(2|h-p) is verified computationally.

s range of L e=5 6 7 8 9 10 11 12
2 2-4 1-1 0-0 1-1 0-0 0-0 0-0 1-1 0-0
3 4-8 1-0 1-1 0-0 0-0 0-0 0-0 0-0 1-1
4 8-16 0-0 1-1 1-1 0-0 1-1 0-0 0-0 0-0
516-32 1-1 0-0 1-0 1-0 1-1 1-1 0-0 1-0
632-64 1-0 0-0 1-1 0-0 1-1 2-1 3-2 0-0
764-27 3-1 2-1 0-0 3-1 1-0 0-0 1-0 1-0
827-28 2-1 3-1 5-2 3-2 1-0 3-2 3-1 4-2
928-29 7-1 7-4 8-1 7-3 4-1 4-3 3-1 2-1
1029-210 9-4 6-0 8-5 12-2 5-3 3-0 7-1 8-2
accumulated total 25-920-825-1126-8 14-713-718-617-6
11210-211 11-3 12-5 13-5 17-6 12-3 15-7 13-4 10-6
12211-212 30-9 20-8 27-14 20-12 20-13 23-11 26-8 22-8
13212-213 46-15 46-21 48-13 41-17 40-15 38-16 38-11 30-12
14213-214 68-23 91-38 76-23 69-30 77-24 61-26 58-19 61-24
15214-215 147-45 144-60 133-36 111-43 135-39 120-58 117-33 105-40
16215-216 254-81 260-100 235-69 220-100 235-58 222-91 216-70 210-93
17216-217 485-140 486-230 424-125 438-191 410-119 382-181 381-113 371-161
18217-218 846-277 834-365 797-239 813-338 712-206 729-329 707-228 697-298
19218-219 1664-536 1514-663 1450-428 1408-601 1339-403 1319-588 1245-353 1156-508
20219-220 2831-888 2758-1216 2697-776 2588-1134 2450-786 2503-1098 2258-647 2280-993
accumulated total 6407-20266185-27145925-17395751-2480 5444-16735425-24125077-14924959-2149
21220-221 5302-1659 5083-2227 4962-1449 4811-2127 4529-1406 4484-2005 4306-1245 4148-1846
22221-222 9724-3052 9401-4151 9011-2616 8644-3827 8527-2689 8312-3726 7943-2300 7691-3327
23222-223 17829-5645 17266-7680 16685-4914 16253-7237 15909-5012 15055-6630 14780-4463 14310-6372
24223-224 33380-10496 32092-14023 30989-9029 29892-13078 29022-9123 28457-12412 27530-8157 26874-11795
25224-225 61492-19489 59346-25958 57737-16856 55945-24659 54425-17135 52678-23022 51519-15036 49961-21759
26225-226 114383-35965 110593-48274 107502-31490 104496-45940 101367-31795 98760-43128 96139-28272 93463-41042
27226-227 213252-67010 207055-90655 200728-58185 195508-86064 189967-59859 184683-81030 179808-52606 175809-76837
28227-228 399050-125720
(935s)		 387730-169521
(1033s)		 376334-109681
(1137s)		 366757-161108
(1393s)		 355797-111640
(1697s)		 347360-152182
(2164s)		 338617-99243
(2710s)		 329938-144180
(3557s)
accumulated total 860819-271062834751-365203809873-235959788057-346520 764987-240332745214-326547725719-212814707153-309307
6,236,573 - 2,307,754
Results
of package
computation
*1		 [File] [File] [File] [File] [File] [File] [File] [File]
*1 : All files in the row are 20-30MB LARGE.

s range of L e=13 14 15 16 17 18 19 20 21 22 23 24
2 2-4 0-0 0-0 0-0 0-0 1-1 0-0 0-0 0-0 0-0 0-0 0-0 0-0
3 4-8 1-0 1-1 0-0 0-0 0-0 0-0 1-0 0-0 0-0 0-0 0-0 1-1
4 8-16 0-0 0-0 0-0 0-0 0-0 1-1 1-1 1-1 0-0 0-0 0-0 0-0
5 16-32 0-0 1-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0
6 32-64 0-0 0-0 1-1 1-1 1-0 1-0 0-0 1-1 1-1 0-0 0-0 0-0
7 64-27 4-1 0-0 1-0 1-1 2-0 1-0 2-1 2-1 0-0 2-1 2-0 0-0
8 27-28 3-1 5-4 0-0 4-1 2-1 4-3 2-1 1-1 2-1 0-0 3-0 2-1
9 28-29 3-2 3-2 5-1 3-1 5-0 2-2 2-0 2-0 1-0 2-1 4-1 2-1
10 29-210 3-0 4-1 7-1 5-1 6-3 5-1 6-2 3-0 5-3 2-1 4-1 2-0
accumulated total 14-4 14-8 14-3 14-5 17-5 14-7 14-5 10-4 9-5 6-3 13-2 7-3
11 210-211 13-2 9-4 9-5 10-4 8-1 12-3 11-2 8-1 10-2 5-1 5-0 12-7
12 211-212 19-5 19-4 21-8 14-5 14-4 24-12 10-3 24-13 13-4 11-3 14-7 15-7
13 212-213 37-12 29-12 21-5 26-13 24-9 33-16 33-12 32-16 25-5 19-8 24-6 23-11
14 213-214 53-11 47-17 54-18 56-27 48-13 45-22 42-8 50-22 56-21 46-23 46-13 44-23
15 214-215 96-29 110-53 83-22 101-46 83-19 99-47 90-23 74-28 88-34 79-39 89-28 76-40
16 215-216 207-65 210-82 168-53 182-77 180-51 161-70 165-47 152-59 145-42 137-60 135-36 138-64
17 216-217 376-119 363-159 327-82 321-146 324-95 321-152 313-100 304-143 285-77 286-135 270-80 259-129
18 217-218 676-204 632-274 644-177 616-259 577-178 610-250 494-163 554-242 471-140 521-207 465-140 468-199
19 218-219 1180-358 1148-511 1114-316 1080-479 1082-323 1004-443 1054-304 1020-451 947-297 959-404 934-277 883-381
20 219-220 2176-661 2090-918 2037-628 2032-896 1900-576 1819-793 1914-550 1753-783 1806-576 1769-772 1731-481 1707-735
accumulated total 4847-1470 4671-2042 4492-1317 4452-1957 4257-1274 4142-1815 4140-1217 3981-1762 3855-1203 3838-1655 3726-1070 3632-1599
21 220-221 4077-1268 3950-1720 3804-1084 3767-1663 3658-1177 3581-1577 3438-1018 3391-1484 3278-1021 3234-1433 3134-924 3036-1329
22 221-222 7496-2330 7271-3146 7158-2110 6981-3097 6704-2167 6583-2903 6387-1862 6244-2661 6173-1951 5980-2635 5771-1699 5765-2576
23 222-223 13999-4353 13473-5815 13231-3809 12994-5655 12522-3895 12422-5428 11963-3511 11689-5100 11320-3501 11260-4944 11013-3245 10673-4727
24 223-224 26111-8202 25437-11371 24724-7176 23975-10539 23532-7402 22882-9987 22353-6613 22194-9635 21490-6616 20909-9157 20485-6023 19983-8787
25 224-225 48525-15038 47620-21153 46162-13357 45093-19776 44194-14058 43114-18839 42351-12496 40929-17807 40039-12488 39344-17258 38804-11275 37747-16644
(22h)
26 225-226 91792-28511 89036-39489 86620-25515 85246-37376 83014-26246 80691-35452 79052-23349 77677-34046 76020-23691 74324-32355 72944-21344
(15h)		 80836-35662
27 226-227 170968-53899 167236-73961 163723-47899 159528-69915 156052-49411 152634-66807 148666-43649 146170-64069 142783-44809 139980-61158
(25h)		 155877-45580
28 227-228 321880-100852
(1.3h)		 314870-138996
(1.9h)		 307590-89874
(2.5h)		 300122-131331
(3.6h)		 294066-92891
(5.1h)		 287297-125708
(7.3h)		 281666-82682
(11h)		 275395-120215
(20h)		 270438-84421
(26h)		 298869-130595
accumulated total 689695-215923 673564-297693 657504-192141 642158-281309 627999-198521 613346-268516 600016-176397 587670-256779 575396-179701
5,665,348 - 2,066,980 535,582 - 211,837
6,200,930 - 2,278,817
Computation
Results		 [File] [File] [File] [File] [File] [File]

s range of L e=25 26 27 28 29 30 31 32 33 34 35 36
2 2-4 0-0 0-0 0-0 0-0 1-1 0-0 0-0 0-0 0-0 0-0 1-1 0-0
3 4-8 1-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0
4 8-16 0-0 0-0 1-1 0-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0
5 16-32 0-0 2-1 0-0 1-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0 0-0
6 32-64 2-1 1-1 0-0 0-0 0-0 0-0 1-0 0-0 0-0 1-1 0-0 0-0
7 64-27 0-0 3-2 0-0 0-0 2-0 0-0 1-0 1-0 0-0 0-0 0-0 1-0
8 27-28 0-0 2-1 2-1 1-0 2-1 1-0 3-2 2-1 1-1 5-4 1-0 0-0
9 28-29 5-2 2-2 3-0 3-1 4-1 1-1 1-0 3-2 3-0 1-0 6-2 0-0
10 29-210 3-1 5-2 4-2 4-2 1-1 5-1 4-0 4-0 3-2 6-4 6-4 3-1
accumulated total 11-415-910-49-3 10-47-210-210-3 7-313-914-74-1
11 210-211 8-1 6-3 5-3 9-4 3-0 5-2 6-2 6-1 7-1 5-4 4-0 6-4
12 211-212 16-6 16-7 12-6 12-4 16-10 18-9 14-6 6-2 14-5 9-2 10-6 11-5
13 212-213 24-8 32-12 22-4 28-9 13-1 19-5 15-2 31-11 19-7 29-14 23-10
(21h)		 14-4
14 213-214 36-9 41-19 36-11 40-19 36-8 40-18 40-7 36-20 22-7 28-15 38-10
(25h)		 35-14
15 214-215 66-20 74-31 80-22 57-24 78-23 67-27 67-13 59-20 54-10 65-28
(38h)		 89-33
16 215-216 132-40 132-63 124-32 128-52 128-34 118-55 106-30 111-46 130-45
(21h)		 149-72
17 216-217 267-86 268-131 243-67 244-115 264-86 213-98 226-72 212-94
(26h)		 253-78
18 217-218 479-153 453-179 447-128 447-187 388-122 436-188 438-134
(17h)		 471-197
19 218-219 868-275 885-359 885-274 840-369 794-269 771-324
(24h)		 922-268
20 219-220 1559-469 1603-680 1520-450 1528-659 1450-434
(16h)		 1694-728
accumulated total 3466-10713525-14933384-10013342-1445 3180-991
21 220-221 3117-977 2888-1292 2843-838 2831-1242
(20h)
22 221-222 5594-1739 5521-2395 5507-1646
(15h)		 6173-2687
23 222-223 10392-3177 10385-4530
(20h)		 11734-3485
24 223-224 19869-6313
(16h)		 22319-9710
accumulated total 42438-13277
82,664 - 29,159 6,267 - 2,184 526 - 197
89,457 - 31,540

s range of L e=37 38 39 40 41 42 43 44 45 46 47
2 2-4 0-0 0-0 0-0 1-1
(49h)		 0-0 0-0 0-0 0-0 0-0 0-0 0-0
3 4-8 0-0 1-1
(16h)		 0-0 0-0 0-0 0-0 0-0 0-0
4 8-16 0-0 0-0 0-0 0-0 0-0 1-1
(24.4days) 		0-0
5 16-32 0-0 0-0 0-0 1-0 0-0 1-1
(25.2days) 		0-0
6 32-64 0-0 0-0 0-0 0-0 0-0 0-0 0-0
7 64-27 0-0 1-1 2-0 0-0 1-0 0-0 0-0
8 27-28 0-0 2-1 0-0 1-0 0-0 2-2
9 28-29 2-1
(6.2h)		 1-1 1-0 1-1
(51h)		 1-0
10 29-210 4-2
(22h)		 4-2
(28h)		 3-1
(23h)		 4-2
accumulated total 6-39-66-1
11 210-211 2-0 4-2
(33h)
12 211-212 4-0 13-8
accumulated total 12-3
38 - 16
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pcf2:
Note on the class number of the p th cyclotomic field, II
Shoichi FUJIMA and Humio ICHIMURA