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Repunit primes
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A repunit (in base 10) is a number like 1, 11, 111, or 1111 that contains only the digit 1 (in its base 10 representation). The term stands for repeated unit and was coined in 1966 by Albert H. Beiler. A repunit prime is a repunit that is also a prime number.
Repunit primes (in base 10) are primes of the form
-
Rk :=
=10 k − 1 10 − 1 k − 1∑ i = 0(10 − 1) 10 i, k ≥ 1.
Generalized repunit primes
A generalized repunit (in base| b |
| b |
| b |
-
R (b) k :=
=b k − 1 b − 1 k − 1∑ i = 0(b − 1) b i, b ≥ 2, k ≥ 1.
| b |
| b |
| b |
| b |
| b |
| b |
Repunit primes in base b: primes of the form (b^p-1)/(b-1) (Verify.) [2]
b values of p
2: 2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281 3217 4253 4423 9689 9941 11213 19937 21701 23209 44497 86243 110503 132049 216091 756839 859433 1257787 1398269 2976221 3021377 6972593 13466917 20996011 24036583 25964951 30402457 32582657 (?) 37156667 (?) 42643801 (?) 43112609 (?) 57885161 (A000043)
3: 3 7 13 71 103 541 1091 1367 1627 4177 9011 9551 36913 43063 49681 57917 483611 877843 (A048491)
4: 2 (No others)
5: 3 7 11 13 47 127 149 181 619 929 3407 10949 13241 13873 16519 201359 396413 (A004061)
6: 2 3 7 29 71 127 271 509 1049 6389 6883 10613 19889 79987 608099 (A004062)
7: 5 13 131 149 1699 14221 35201 126037 371669 1264699 (A004063)
8: 3 (No others)
9: (None)
10: 2 19 23 317 1031 49081 86453 109297 270343 (A004023)
11: 17 19 73 139 907 1907 2029 4801 5153 10867 20161 293831 (A005808)
12: 2 3 5 19 97 109 317 353 701 9739 14951 37573 46889 769543 (A004064)
13: 5 7 137 283 883 991 1021 1193 3671 18743 31751 101089 (A016054)
14: 3 7 19 31 41 2687 19697 59693 67421 441697 (A006032)
15: 3 43 73 487 2579 8741 37441 89009 505117 (A006033)
16: 2 (No others)
17: 3 5 7 11 47 71 419 4799 35149 54919 74509 (A006034)
18: 2 25667 28807 142031 157051 180181 414269 (A133857)
19: 19 31 47 59 61 107 337 1061 9511 22051 209359 (A006035)
20: 3 11 17 1487 31013 48859 61403 472709 (A127995)
21: 3 11 17 43 271 156217 328129 (A127996)
22: 2 5 79 101 359 857 4463 9029 27823 (A127997)
23: 5 3181 61441 91943 (A204940)
24: 3 5 19 53 71 653 661 10343 49307 (A127998)
25: (None)
26: 7 43 347 12421 12473 26717 (A127999)
27: 3 (No others)
28: 2 5 17 457 1423 (A128000)
29: 5 151 3719 49211 77237 (A181979)
30: 2 5 11 163 569 1789 8447 72871 78857 82883 (A098438)
31: 7 17 31 5581 9973 54493 101111 (A128002)
32: (None)
33: 3 197 3581 6871 (A209120)
34: 13 1493 5851 6379 (A185073)
35: 313 1297
36: 2 (No others)
37: 13 71 181 251 463 521 7321 36473 48157 87421 (A128003)
38: 3 7 401 449 (A128004)
39: 349 631 4493 16633 36341 (A181987)
40: 2 5 7 19 23 29 541 751 1277 (A128005)
41: 3 83 269 409 1759 11731 (A239637)
42: 2 1319
43: 5 13 6277 26777 27299 40031 44773 (A240765)
44: 5 31 167
45: 19 53 167 3319 11257 (A242797)
46: 2 7 19 67 211 433 2437 2719 19531 (A243279)
47: 127 18013
48: 19 269 349 383 1303 15031 (A245237)
49: (None)
50: 3 5 127 139 347 661 2203 6521 (A245442)
51: 4229
52: 2 103 257 4229 6599
53: 11 31 41 1571 (A173767)
54: 3 389 16481 18371
55: 17 41 47 151 839 2267 3323 3631 5657
56: 7 157 2083 2389
57: 3 17 109 151 211 661 16963
58: 2 41 2333
59: 3 13 479 12251
60: 2 7 11 53 173
61: 7 37 107 769
62: 3 5 17 47 163 173 757 4567 9221 10889
63: 5 3067
64: (None)
65: 19 29 631
66: 2 3 7 19 19973
67: 19 367 1487 3347 4451 10391 13411
68: 5 7 107 149 2767
69: 3 61 2371 3557 8293
70: 2 29 59 541 761 1013 11621
71: 3 31 41 157 1583
72: 2 7 13 109 227
73: 5 7
74: 5 191 3257
75: 3 19 47 73 739 13163 15607
76: 41 157 439 593 3371 3413 4549
77: 3 5 37 15361
78: 2 3 101 257 1949
79: 5 109 149 659
80: 3 7
81: (None)
82: 2 23 31 41 7607 12967
83: 5 2713
84: 17 3917
85: 5 19 2111
86: 11 43 113 509 1069 2909 4327
87: 7 17
88: 2 61 577 3727
89: 3 7 43 47 71 109 571 11971
90: 3 19 97 5209
91: 4421
92: 439 13001
93: 7 4903
94: 5 13 37 1789 3581
95: 7 523 9283 10487 11483
96: 2 3343
97: 17 37 1693
98: 13 47 2801
99: 3 5 37 47 383 5563
100: 2 (No others)
101: 3 337 677 1181 6599
102: 2 59 673
103: 19 313 1549
104: 97 263 5437
105: 3 19 389 2687 4783
106: 2 149
107: 17
108: 2 449 2477
109: 17 1193 13679
110: 3 5 13 691 1721 3313 11827
111: 3 337
112: 2 79 107 701 1697 5657
113: 23 37 6563
114: 29 43 73 89 569 709
115: 7 241 1409 2341 2539 7673 12539 16879
116: 59 2503
117: 3 5 19 31
118: 5 163 193
119: 3 19 827 2243 3821
120: 5 373 1693
121: (None)
122: 5 7 67 3803
123: 43 563 1693 4877
124: 599 18367
125: (None)
126: 2 7 37 59 127
127: 5 23 31 167 5281 8969 23297
128: 7 (No others)
See also
Notes
- ↑ 1.0 1.1 Harvey Dubner, Generalized Repunit Primes, Mathematics of Computation, 61, 204, Oct., 1993.
- ↑ Needs verification.
