Talk:Repunit

Every repunit is a repdigit. --Abdull 20:08, 19 March 2006 (UTC)

The sum of the reciprocals of the repunit primes converges. This result could suggest the finiteness of Repunit primes, just like Brun's theorem could suggest the finiteness of twins.

:The sum of the reciprocals of all repunit numbers also converges, but there are infinitely many repunit numbers. This says nothing about infinitude of repunit primes. Standard heuristics suggest there are probably infinitely many. PrimeHunter 12:48, 23 January 2007 (UTC)

Let the numbers of repunit primes be finite. Then, the sum of the reciprocals of the repunit primes diverges if there are infinitely many repunit primes. 218.133.184.93 08:57, 15 February 2007 (UTC)

:No. As I said above, the sum of the reciprocals of all repunit numbers converges. Only some of the repunit numbers are repunit primes, so the sum of reciprocals of repunit primes is smaller. It converges whether there are infinitely many or not. PrimeHunter 11:59, 15 February 2007 (UTC)

:Yes. Anything is true when the premise is false.218.133.184.93 04:42, 16 February 2007 (UTC)

::The false premise is that 218.133.184.93's statement is relevant. — Arthur Rubin | (talk) 23:32, 13 August 2007 (UTC)

::The false premise is that Arthur Rubin is busy.218.133.184.93 01:33, 16 August 2007 (UTC)

It says that repunits are also prime numbers. Doesn't 111 easily break that rule? 37*3=111. Therefore, no longer prime. Wikifor (talk) 06:38, 6 February 2010 (UTC)

:The article says "A repunit prime is a repunit that is also a prime number" and later "Only repunits (in any base) having a prime number of digits might be prime (necessary but not sufficient condition.)". Some repunits are prime but most are not. 111 is not. PrimeHunter (talk) 12:18, 6 February 2010 (UTC)

"Except for this case of R_3, p can only divide R_n if p = 2kn + 1 for some k."

How does this fit to 11 dividing every R_2n? --91.13.253.19 (talk) 22:46, 7 December 2007 (UTC)

:Prime n had just been discussed and the quoted statement assumes n is prime. I have added it to the article for clarity.[http://en.wikipedia.org/w/index.php?title=Repunit&diff=176472592&oldid=176467057] PrimeHunter (talk) 00:09, 8 December 2007 (UTC)

"It is easy to show that if n is divisible by a, then R_n is divisible by R_a:"

Wouldn't this mean that R(ab) = R(a) * R(b) (which is obviously not true)? 213.216.248.212 (talk) 10:52, 29 October 2008 (UTC)

:No, it would mean that R(ab) is divisible by both R(a) and R(b). That doesn't mean it's equal to the product. For example, R(6) = R(2) * R(3) * 91. But R(ab) doesn't even have to be divisible by the product when a and b have a common factor. For example, R(4) = 1111 = 11 * 101 is divisible by R(2) but not by R(2)*R(2). PrimeHunter (talk) 11:36, 29 October 2008 (UTC)

Hi All,

As the current world record holder for proving prime generalized repunits and maintainer of a page listing all known such primes, it would not be proper for me to edit the wiki page. In case anyone _else_ thinks my page would be a useful link, here it is:

http://www.primes.viner-steward.org/andy/titans.html

Also, there is a Repunit Primes collaborative project in progress at http://www.gruppoeratostene.com/ric-repunit/repunit.htm

Cheers,

Andy Steward

88.106.202.253 (talk) 14:12, 15 March 2010 (UTC)

Should we perhaps have a seperate article about repunit primes (like the way we have an article about Mersenne numbers and Mersenne primes)? Or should we really just keep all information about repunit primes in this article? Toshio Yamaguchi (talk) 16:26, 30 October 2010 (UTC)

:Repunits are generally only studied for the sake of finding their factors, so I don't think a split is needed. In any case Mersenne number redirects to Mersenne prime; it has never been a separate article. Xanthoxyl ^< 08:05, 31 October 2010 (UTC)

::Ok agreed. Regarding the Mersenne number article I must have missed something, sorry. Toshio Yamaguchi (talk) 11:27, 31 October 2010 (UTC)

Currently, the article lists some larger numbers which are repunits in a specific base. I must question the usefulness of listing the whole string of digits of these numbers in the article. Wouldn't it be better to use some shortened form of notation? I think listing those large numbers is more distracting than useful to the reader. Toshio Yamaguchi (talk) 10:51, 12 March 2011 (UTC)

:Listing the first few is fine. It's not like people are going to injure themselves with the long pointy numbers. And as for the "usefulness"... what use are prime repunits? Xanthoxyl ^< 11:54, 12 March 2011 (UTC)

::I agree that it doesn't hurt, as long as the string of digits doesn't exceed a particular lenght. But at some point, listing them becomes distracting. For example, seeing the string of digits of {{nowrap|138502212710103408700774381033135503926663324993317631729227790657325163310341833227775945426052637092067324133850503035623601}} doesn't really impart any useful information to a reader and only clutters the article. Therefore I think using some kind of short notation would be more useful. If the string of digits is really important, an external site showing the digits should be given in the external links. Toshio Yamaguchi (talk) 12:17, 12 March 2011 (UTC)

:::If the reader were forced to scroll past oceans of digits, that would be one thing, but we're just talking about a couple of lines. I think that a glimpse of the size of these numbers is more likely to pique the reader's interest than put them off. (And again: useful? to whom are repunits useful?) Xanthoxyl ^< 12:47, 12 March 2011 (UTC)

::::I didn't question the usefulness on having an article about Repunits. Usefulness is not an argument for or against inclusion of a topic in Wikipedia, only notability is. And your argument "We're just talking about a couple of lines" isn't an argument either. I also don't see the benefit of showing the 'size' of these numbers by listing their digits in the article. A short notation that is explained in an understandable way would do it. Toshio Yamaguchi (talk) 13:34, 12 March 2011 (UTC)

:::::I'm sorry, did you actually have an argument against inclusion? If so, I didn't notice it. You said "it should be removed because it's not useful" and then you immediately said "usefulness is not an argument". I'm not sure why a couple of long numbers should cause you such psychological distress, but by all means delete them if you believe the sight of 100 digits could induce seizures or uncontrollable panic. Xanthoxyl ^< 14:24, 12 March 2011 (UTC)

:The 127-digit base-7

138502212710103408700774381033135503926663324993317631729227790657325163310341833227775945426052637092067324133850503035623601

:goes off the right of my screen, and I'm sure many other users. I suggest a cut-off at 110 digits to include the 110-digit base-7

85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457

:There are known cases with thousands of digits so we have to exclude some of them. Currently we omit an 88-digit and 104-digit base-5 case. They would be added under my suggestion. I also suggest we list the n values for more primes which are too large for the decimal expansions. PrimeHunter (talk) 14:55, 12 March 2011 (UTC)

::I think we shouldn't list much more than 100 digits but the question is, where exactly to draw the line. Is it really a good idea to determine the upper limit in length by which numbers are included of we choose a particular limit? In this way this limit becomes rather arbitrary as anybody could argue "I want to have number xy included, thus the limit should be z". The limit should be chosen by which numbers can be presented while still giving useful information. (And I think presenting any numbers with more than 100 digits doesn't any longer give any useful information). Toshio Yamaguchi (talk) 15:45, 12 March 2011 (UTC)

:@Xanthoxyl: So if you think 138502212710103409700774381033135503926663324993317631729227790657325163310341833227775945426052637092067324133850503035623601

:should be in the article, just let me say that I exchanged the 18th digit by a 9 (it was 8 before). Would you have recognized, if I hadn't told you? Probably not. Thus I think we should only show numbers in full lenght, if they can be easily recognized as the number in question. For example, if in the article Wilson prime someone changed 563 to 593, that could be easily recognized. But in the example above, this number is hardly distinguishible from most other numbers of that lenght. Therefore, if the digits are important (for example to paste the number into a factoring program), an external link to a source showing the digits should be provided. Toshio Yamaguchi (talk) 19:04, 12 March 2011 (UTC)

::1) This argument, if taken seriously, would leave Wikipedia with no tables or dates or hard information of any sort. The wiki format is set up expressly to detect these sorts of alterations.

::2) Precedent shows [http://en.wikipedia.org/wiki/Talk:Pascal's_triangle#Modifyed_triangle_.22vandalism.22] that the shortness of an erroneous number is no guarantee that it will be detected quickly. In one section of Pascal's triangle, the incorrect "104" was written by accident on 15 April and not noticed until 21 October 2005.

::3) No one has pointed to a policy, and I find that there are no guidelines on WP:MOSMATH or WP:MOSNUM. I would have suggested a maximum of 100 digits for the largest number in a list or table, and a maximum of 40 digits for a number appearing in a sentence. But as always, the standard is common sense: does the inclusion of the data irk the average reader, unbalance the page, or push more important information out of the way? Xanthoxyl ^< 04:08, 13 March 2011 (UTC)

The only guideline which might be applicable here seems to be Wikipedia:Manual of Style (dates and numbers)#Large numbers. It is not very clear on this case and only says: "Scientific notation is preferred in scientific contexts". Thus I think some form of short scientific notation should be used for the larger numbers, like R₁₃₁ and R₁₄₉ for the 3rd and 4th base 7 repunits respectively. But this is only my opinion and I think a consensus regarding this matter should be reached. I would like to invite all interested editors to express their opinions in order to reach a consensus. Toshio Yamaguchi (talk) 18:24, 13 March 2011 (UTC)

The article mentions many repunit numbers in different bases, but then lists the numbers in (presumably) decimal. However, it does look extremely odd to read, for example: "The first few base-3 repunit primes are 13, 1093, 797161, ..." when the reader surely knows 0, 1 & 2 are the only valid digits in base-3. I would have expected to read: "The first few base-3 repunit primes are 111, 1111111, 1111111111111, ..." Astronaut (talk) 13:21, 14 March 2011 (UTC)

:Rather than force the reader to sit counting a string of ones, I'd suggest inserting the words "in decimal". Xanthoxyl ^< 15:50, 14 March 2011 (UTC)

::Per WP:ORDINAL we could write 111₃, 1111111₃, 1111111111111₃, ... but again the question is up to which lenght (number of ones) does this notation make sense? Toshio Yamaguchi (talk) 16:35, 14 March 2011 (UTC)

If the reference to the binomial theorem was the primary objection to the proof of the "holographic repdigits" theorem and cause of such nonconstructive, unexplained reversion, I am nonplussed. The binomial theorem is of course not essential to the proof, but McGough and Curfs include it. I assume they actually constructed some binomial expansions in Z and used that theorem to reassure that d divided every term except the last. The proof does not become incorrect somehow if it is included.

Reverting with such poorly explained reasons is stupid and unhelpful. If you believe an error exists, correct it or state where it is. — Preceding unsigned comment added by 64.134.229.15 (talk) 02:45, 26 April 2013 (UTC)

:We're dealing with the theorem that if

::$a\; \backslash equiv\; b\; \backslash pmod\; n,$

:then

::$a^N\; \backslash equiv\; b^N\; \backslash pmod\; n.$

:As that's used implictly elsewhere in the argument, there's no point in using it explictly, binomial theorem or not. — Arthur Rubin (talk) 04:09, 26 April 2013 (UTC)

What we need is WikiTable or HTML formatting elements to force (1) each entry to be at the top of the cell, rather than centered vertically, (2) (Left, rather than centered horizontally, which seems already to have been done), and (3) each R_n to be in line with the "=". The anon's recent addition of improper HTML
s doesn't really solve the problem, and doesn't maintain proper alignment. — Arthur Rubin (talk) 18:13, 21 August 2014 (UTC)

These are generalized repunit prime in base 2 to 257 and −2 to −257 and some other bases.

Positive bases (2 to 257):

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 (?) 74207281 (?) 77232917

3 3 7 13 71 103 541 1091 1367 1627 4177 9011 9551 36913 43063 49681 57917 483611 877843

4 2 (no others)

5 3 7 11 13 47 127 149 181 619 929 3407 10949 13241 13873 16519 201359 396413

6 2 3 7 29 71 127 271 509 1049 6389 6883 10613 19889 79987 608099

7 5 13 131 149 1699 14221 35201 126037 371669 1264699

8 3 (no others)

9 (none)

10 2 19 23 317 1031 49081 86453 109297 270343

11 17 19 73 139 907 1907 2029 4801 5153 10867 20161 293831

12 2 3 5 19 97 109 317 353 701 9739 14951 37573 46889 769543

13 5 7 137 283 883 991 1021 1193 3671 18743 31751 101089

14 3 7 19 31 41 2687 19697 59693 67421 441697

15 3 43 73 487 2579 8741 37441 89009 505117

16 2 (no others)

17 3 5 7 11 47 71 419 4799 35149 54919 74509

18 2 25667 28807 142031 157051 180181 414269

19 19 31 47 59 61 107 337 1061 9511 22051 209359

20 3 11 17 1487 31013 48859 61403 472709

21 3 11 17 43 271 156217 328129

22 2 5 79 101 359 857 4463 9029 27823

23 5 3181 61441 91943 121949

24 3 5 19 53 71 653 661 10343 49307 115597

25 (none)

26 7 43 347 12421 12473 26717

27 3 (no others)

28 2 5 17 457 1423

29 5 151 3719 49211 77237

30 2 5 11 163 569 1789 8447 72871 78857 82883

31 7 17 31 5581 9973 54493 101111

32 (none)

33 3 197 3581 6871

34 13 1493 5851 6379

35 313 1297

36 2 (no others)

37 13 71 181 251 463 521 7321 36473 48157 87421

38 3 7 401 449 109037

39 349 631 4493 16633 36341

40 2 5 7 19 23 29 541 751 1277

41 3 83 269 409 1759 11731

42 2 1319

43 5 13 6277 26777 27299 40031 44773

44 5 31 167 100511

45 19 53 167 3319 11257 34351

46 2 7 19 67 211 433 2437 2719 19531

47 127 18013 39623

48 19 269 349 383 1303 15031

49 (none)

50 3 5 127 139 347 661 2203 6521

51 4229 35227

52 2 103 257 4229 6599

53 11 31 41 1571 25771

54 3 389 16481 18371 82471

55 17 41 47 151 839 2267 3323 3631 5657 35543

56 7 157 2083 2389 57787

57 3 17 109 151 211 661 16963 22037

58 2 41 2333 67853

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 38609

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 106397

70 2 29 59 541 761 1013 11621 27631

71 3 31 41 157 1583 31079 55079 72043

72 2 7 13 109 227

73 5 7 35401

74 5 191 3257 31267

75 3 19 47 73 739 13163 15607 93307

76 41 157 439 593 3371 3413 4549

77 3 5 37 15361

78 2 3 101 257 1949 67141

79 5 109 149 659 28621

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 40583

87 7 17

88 2 61 577 3727 22811 40751

89 3 7 43 47 71 109 571 11971 50069

90 3 19 97 5209

91 4421 20149

92 439 13001 22669 44491

93 7 4903

94 5 13 37 1789 3581

95 7 523 9283 10487 11483

96 2 3343 46831

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 25087

103 19 313 1549

104 97 263 5437

105 3 19 389 2687 4783

106 2 149

107 17 24251

108 2 449 2477

109 17 1193 13679 27061

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 22741

124 599 18367 28591

125 (none)

126 2 7 37 59 127 20947

127 5 23 31 167 5281 8969 23297 165601

128 7 (no others)

129 5 17 109 8447

130 2 37

131 3 31 263

132 47 71 3343

133 13 599 991 1181 3083 14827

134 5 37 353 2843 21379

135 1171 15227

136 2 227 293 4133

137 11 19 1009 2939

138 2 3 61 13679

139 163 173 3821

140 79 577 1721

141 3 23 173 3217

142 1231 6133

143 3 5

144 (none)

145 5 31

146 7 83 857 21961

147 3 17 19 37 163 571 983 3697

148 2 1201

149 7 13 17 317 3251

150 2 3 3389

151 13 29 127 4831 5051 13249 18251

152 270217

153 3 5099

154 5 8161

155 3 61 449 2087

156 2 7 199 5591

157 17 107 2791 39047 53819 90239

158 7 79 109 4003 6151 10453

159 13 89 577 1433 9643

160 7 17 151 1487 3989

161 3 37 263

162 2 3 5 311 1087

163 7 43 241 1637 2543

164 3 5 19 101 347 383

165 5 53 109

166 2 137 353 1289

167 3 19 373 1213 2203

168 3 823

169 (none)

170 17 23 79 1237 19843

171 181 3373 12391

172 2 5 11 37 47

173 3 2687

174 3251

175 5 167 1699 5881

176 3 151 2719 3923 11743 13397

177 5 31

178 2 347 911 4523

179 19

180 2 7 43 1913 2683 4637

181 17 19 157

182 167 509 1609

183 223

184 16703

185

186 7 47 223 271 3947 4153 10177

187 37 617

188 3 59 3719

189 3 17

190 2 13 89 157 643 673 10427

191 17 1399

192 2 3 7 613

193 5 317 11171

194 3 8807

195 11 73 379 2687

196 2 (no others)

197 31 47 283 11719

198 2 5 9721 10771

199 577 1831

200 17807

201 271 353

202 37 829

203 3 7 31 9587

204 5 359

205 19 61 6427 8147

206 3 7

207 13 17

208 5 7 37 1229 1583 3517

209 3 59 449 613

210 2 19819

211 41

212 11

213 137

214 191

215 3 73 461 751 3433

216 (none)

217 281 821

218 3 331 701 971 1277

219 13 107 223 1307

220 7 19 47 307

221 7 13 29 139 223 439 6907

222 2 5 151 271 5077

223 239 241 449

224 11 401

225 (none)

226 2 127 619 7043

227 5 1061 2687

228 2 461 4801 11443

229 11 29

230 5333

231 3 6907

232 2 953 2801 4111

233 113 9511

234 61 89 97 1381 9011

235 7 19 53 227 307

236 3 197 467 587

237 7 2621

238 2 7 67 1093 1381

239 5 109 2549

240 2 109 227 271 941

241 17 31

242 19 541

243 (none)

244 3331 5099

245 3 9277

246 3 37 251

247 17 331

248 41 197 2203

249 5 1249 2053 3319 8627

250 2 127 1889

251 7 13 17 89 227 461 3467

252 541 947

253 19 2659

254 5 19 79 283 563 883

255 5 151 701

256 2 (no others)

257 23 59 487 967 5657

Positive bases (some special bases > 257):

290 = 17²+1 3 7

325 = 18²+1 31 1039

344 = 7³+1 3 23

362 = 19²+1 199 2663

401 = 20²+1 127 199 6551

442 = 21²+1 2 13 23 199 5309

485 = 22²+1

511 = 2⁹−1

513 = 2⁹+1 17 2663 6883

530 = 23²+1 3 5 599

577 = 24²+1 109 139 227

626 = 5⁴+1 3 11 61 1249

730 = 3⁶+1 13

1001 = 10³+1 3 1787

1023 = 2¹⁰−1 19

1025 = 2¹⁰+1 13 83

1297 = 6⁴+1 5 7 29 2423

1332 = 11³+1 17 3701

1729 = 12³+1 1097

2047 = 2¹¹−1 877

2049 = 2¹¹+1 3 17

2188 = 3⁷+1 7 3011 12437

2402 = 7⁴+1 3 19

3126 = 5⁵+1 11 2749 14431 14983

4095 = 2¹²−1 5479

4097 = 2¹²+1 7 37 3673 8311

6562 = 3⁸+1 2 701

7777 = 6⁵+1 5

10001 = 10⁴+1 11 569

14642 = 11⁴+1 3

20737 = 12⁴+1 227

65535 = 2¹⁶−1

65537 = 2¹⁶+1 7 11

100001 = 10⁵+1 31 53

1000001 = 10⁶+1 11 277

Positive bases (perfect powers between 257 and 4096 and some other perfect powers):

289 = 17² (none)

324 = 18² (none)

343 = 7³ (none)

361 = 19² (none)

400 = 20² 2 (no others)

441 = 21² (none)

484 = 22² (none)

512 = 2⁹ 3 (no others)

529 = 23² (none)

576 = 24² 2 (no others)

625 = 5⁴ (none)

676 = 26² 2 (no others)

729 = 3⁶ (none)

784 = 28² (none)

841 = 29² (none)

900 = 30² (none)

961 = 31² (none)

1000 = 10³ (none)

1024 = 2¹⁰ (none)

1089 = 33² (none)

1156 = 34² (none)

1225 = 35² (none)

1296 = 6⁴ 2 (no others)

1331 = 11³ 3 (no others)

1369 = 37² (none)

1444 = 38² (none)

1521 = 39² (none)

1600 = 40² 2 (no others)

1681 = 41² (none)

1728 = 12³ (none)

1764 = 42² (none)

1849 = 43² (none)

1936 = 44² (none)

2025 = 45² (none)

2048 = 2¹¹ (none)

2116 = 46² (none)

2187 = 3⁷ (none)

2197 = 13³ (none)

2209 = 47² (none)

2304 = 48² (none)

2401 = 7⁴ (none)

2500 = 50² (none)

2601 = 51² (none)

2704 = 52² (none)

2744 = 14³ (none)

2809 = 53² (none)

2916 = 54² 2 (no others)

3025 = 55² (none)

3125 = 5⁵ (none)

3136 = 56² 2 (no others)

3249 = 57² (none)

3364 = 58² (none)

3375 = 15³ (none)

3481 = 59² (none)

3600 = 60² (none)

3721 = 61² (none)

3844 = 62² (none)

3969 = 63² (none)

4096 = 2¹² (none)

8192 = 2¹³ (none)

16384 = 2¹⁴ (none)

32768 = 2¹⁵ (none)

65536 = 2¹⁶ 2 (no others)

Positive bases (the special cases):

N 2 3 19 31 7547 ==> (N^N-1)/(N-1)

Negative bases (2 to 257):

2 3 4 5 7 11 13 17 19 23 31 43 61 79 101 127 167 191 199 313 347

701 1709 2617 3539 5807 10501 10691 11279 12391 14479 42737 83339 95369 117239 127031 138937 141079

267017 269987 374321 986191 4031399 (?) 13347311 13372531

3 2 3 5 7 13 23 43 281 359 487 577 1579 1663 1741 3191 9209 11257 12743 13093 17027 26633

104243 134227 152287 700897 1205459

4 2 3 (no others)

5 5 67 101 103 229 347 4013 23297 30133 177337 193939 266863 277183 335429

6 2 3 11 31 43 47 59 107 811 2819 4817 9601 33581 38447 41341 131891 196337

7 3 17 23 29 47 61 1619 18251 106187 201653

8 2 (no others)

9 3 59 223 547 773 1009 1823 3803 49223 193247 703393

10 5 7 19 31 53 67 293 641 2137 3011 268207

11 5 7 179 229 439 557 6113 223999 327001

12 2 5 11 109 193 1483 11353 21419 21911 24071 106859 139739

13 3 11 17 19 919 1151 2791 9323 56333 1199467

14 2 7 53 503 1229 22637 1091401

15 3 7 29 1091 2423 54449 67489 551927

16 3 5 7 23 37 89 149 173 251 307 317 30197 1025393

17 7 17 23 47 967 6653 8297 41221 113621 233689 348259

18 2 3 7 23 73 733 941 1097 1933 4651 481147

19 17 37 157 163 631 7351 26183 30713 41201 77951 476929

20 2 5 79 89 709 797 1163 6971 140053 177967 393257

21 3 5 7 13 37 347 17597 59183 80761 210599

22 3 5 13 43 79 101 107 227 353 7393 50287

23 11 13 67 109 331 587 24071 29881 44053

24 2 7 11 19 2207 2477 4951

25 3 7 23 29 59 1249 1709 1823 1931 3433 8863 43201 78707

26 11 109 227 277 347 857 2297 9043

27 (none)

28 3 19 373 419 491 1031 83497

29 7 112153 151153

30 2 139 173 547 829 2087 2719 3109 10159 56543 80599

31 109 461 1061 50777

32 2 (no others)

33 5 67 157 12211

34 3

35 11 13 79 127 503 617 709 857 1499 3823

36 31 191 257 367 3061 110503

37 5 7 2707

38 2 5 167 1063 1597 2749 3373 13691 83891

39 3 13 149 15377

40 53 67 1217 5867 6143 11681 29959

41 17 691

42 2 3 709 1637 17911

43 5 7 19 251 277 383 503 3019 4517 9967 29573

44 2 7 41233

45 103 157 37159

46 7 23 59 71 107 223 331 2207 6841 94841

47 5 19 23 79 1783 7681

48 2 5 17 131 84589

49 7 19 37 83 1481 12527 20149

50 1153 26903 56597

51 3 149 3253

52 7 163 197 223 467 5281 52901 85259

53 21943 24697

54 2 7 19 67 197 991

55 3 5 179 229 1129 1321 2251 15061

56 37 107 1063 4019

57 53 227 18211 20231 22973 87719 111119

58 3 17 1447 11003

59 17 43 991 33613

60 2 3 937 1667 3917 18077 31393

61 7 41 359 17657

62 2 11 29 167 313 16567 38699

63 3 37 41 2131 4027 22283 51439 102103

64 (none)

65 19 31

66 7 17 211 643 28921 58741 63079 67349

67 3 2347 2909 3203

68 2 757 773 71713

69 11 211 239 389 503 4649 24847

70 3 61 97 13399 42737

71 5 37 5351 7499 68539 77761

72 2 3 7 79 277 3119

73 7 39181

74 2 13 31 37 109 17383

75 5 83 6211

76 3 5 191 269 23557

77 37 317

78 3 7 31 661 4217

79 3 107 457 491 2011

80 2 5 13 227 439

81 3 5 701 829 1031 1033 7229 19463

82 293 1279 97151

83 19 31 37 43 421 547 3037 8839

84 2 7 13 139 359 971 1087 3527

85 167 3533 48677

86 7 17 397 7159

87 7 467 43189

88 709 1373 61751

89 13 59 137 1103 4423 82609 101363

90 2 3 47

91 3 11 43 397 21529 37507 61879

92 37 59 113

93 89 571 601 3877

94 71 307 613 1787 3793 10391

95 43 93377

96 37 103 131 263 32369

97

98 2 19 101

99 7 37 41 71

100 3 293 461 11867 90089

101 7 229

102 2 3

103

104 2 673 839 1031

105 11 149 1187 1627

106 3 7 19 23 31 3989

107 103 983 18049

108 2 13 223 15731

109 59 79 811

110 2 23 101 17041

111 3 5 23 53 383 2039 12109

112 3

113

114 2 7 13 1801 12487

115 7 31 293

116 113 1481 2089 16889

117 271

118 3 23 109 2357

119 29 53 797 11491

120 3 31 43 263 4919

121 5 13 97 1499 11321

122 293 3877 12889 22277

123 29 739

124 16427

125 (none)

126 5 13 47 163 239 4523

127 317 1061 23887

128 2 7 (no others)

129 17 227 1753

130 467

131 5 101 3389 3581

132 2 3 101 157 1303

133 5 7 17 59 79 157

134 13 1171 6733

135 5 7 2671 11953

136 5 7 23 59 199 2053 6067

137 101 241 353 1999 21851

138 2 103 577 10781

139 3 17 47 2683 2719

140 2 59

141 5 1471

142 3 7537

143 7 17 19 47 103 4423 18287

144 3 23 41 317 3371

145 7 23 281

146 17 1439 11027

147 11 151 6599

148 3 7 31 43 163 317 1933 5669 11789 19289 22171

149 17 769

150 2 6883 15139

151 3 367 3203 7993 10273 14437

152 2 13 19

153 13 1063 5749

154 3 29 263 601 619 809 1217 2267

155 5

156 3 1301

157 5 157 809 1861 2203

158 2 5 769 5023

159 283 449 1949 7457

160 11 37 1907 10487

161 31 331 1483

162 3 1823 7703

163 3 11 31 661 1999 4079 6917

164 2 7 103 541 1109

165 3 5 383

166 17 5437

167 17 59 1301 3167

168 2 3 31 1741 2099

169 3 7 109 21943

170 7

171 13 149 257 4967

172 37 283 647 4483 5417

173 7 59 569 2647

174 2 3 3191

175 31627

176 5 31 269 479 599 809 1307

177 3 5 19 419

178 61 167 227

179 827 5011 8867

180 2 5 13 7369 8101

181 449 2687 4877

182 2 1487 8081

183 11 16363

184 19 79 149 7283

185 11

186

187

188 22037

189 3 31 71 8123

190 3 19 1153

191 479 1163

192 2 109 197 587 727 1997 2441

193 3 11 67 3253

194 2 19 31

195 3 13 19 43 89 1087 1949 2939

196 43 1049 5441 18089

197 31 37 101 163

198 2 37 151 937

199 313 2579 5387

200 2 7 277

201 43 587 593 2861 7841

202 229

203 5 439

204 3 13 1693 11329

205 5449

206 101 1069

207 3 199

208 61

209 311 433 883

210 3 8311

211 79 6659

212 2 101

213 59 239 6607 7177

214 73 157 8867

215 277

216 3 (no others)

217 499 5981

218 241 2417

219 3 7 251 709 1097

220

221 149

222 1657 2963 4231

223 5 103 857 997 5923

224 2 7 5189

225 383 1277

226 7 71 79 1459 1669 2887 5503

227 89

228 2 7 4241

229 11 1117 4159

230 2 13 23 37 41 313

231 7 17 1217 4643

232 3 11 283 7159

233 11

234 2 7 3253 6211

235 223 1993 6043 9137

236 11 71 149 827 1741

237 3 8677

238 23 353

239 59 601 8867

240 2 7

241 19 37 853 3169 7507

242 2 5 137 2011

243 (none)

244 71 613

245 5 29 547 7207 8731

246 3 227 5897

247 3 43 1993 2801

248 7 11 163 1951 2897 3391

249 19 103 317

250 857 1061 1373 3637

251 5 61

252 2 43 521

253 5 383 1049

254 569 2797

255 7 59 179 263 4283 15527

256 5 13 23029 50627 51479 72337

257 5 47 2909 8747

Negative bases (some squares > 257 and some other bases):

289 = 17² 3 179 181 683

324 = 18² (none)

361 = 19² 5 23 223 4441

400 = 20² 263

441 = 21² 101 197

484 = 22² 257

529 = 23² 587 683

576 = 24² 379 461 1861

625 = 5⁴ 3 7 11 31 67 9173 17737

1296 = 6⁴ 3 2153 3517

2401 = 7⁴ 37 3583 8059

6561 = 3⁸ 19 29 11213

10000 = 10⁴ 3 283 1087

14641 = 11⁴ 13 211

20736 = 12⁴ 7 593

65536 = 2¹⁶ 239

65537 = 2¹⁶+1 5

2³² 3 13619

Negative bases (some perfect powers between 257 and 100000):

343 = 7³ 3 (no others)

512 = 2⁹ (none)

729 = 9³ 3 (no others)

1000 = 10³ (none)

1024 = 4⁵ (none)

1331 = 11³ (none)

1728 = 12³ (none)

2048 = 2¹¹ (none)

2187 = 3⁷ (none)

2197 = 13³ (none)

2500 = 4×5⁴ (none)

2744 = 14³ (none)

3125 = 5⁵ (none)

3375 = 15³ (none)

4096 = 16³ (none)

4913 = 17³ (none)

5184 = 4×6⁴ (none)

5832 = 18³ (none)

6859 = 19³ (none)

7776 = 6⁵ 5 (no others)

8000 = 20³ (none)

8192 = 2¹³ 2 (no others)

9261 = 21³ (none)

9604 = 4×7⁴ (none)

10648 = 22³ (none)

12167 = 23³ (none)

13824 = 24³ (none)

15625 = 25³ (none)

16384 = 4⁷ (none)

16807 = 7⁵ 5 (no others)

19683 = 3⁹ (none)

26244 = 4×9⁴ (none)

32768 = 2¹⁵ (none)

40000 = 4×10⁴ (none)

46656 = 36³ (none)

58564 = 4×11⁴ (none)

59049 = 9⁵ (none)

78125 = 5⁷ (none)

82944 = 4×12⁴ (none)

100000 = 10⁵ (none)

Negative bases (the special cases):

N 3 5 17 157 ==> (N^N+1)/(N+1)

N^2 3 7 29 41 43 61 577 ==> (N^2N+1)/(N^2+1)

— Preceding unsigned comment added by 49.216.117.39 (talk) 13:50, 22 August 2015 (UTC)

:Is there a source for this? It looks like original research. Gap9551 (talk) 17:57, 17 December 2015 (UTC)

"The only base 4 repunit prime is 5 ($11\_4$). $4^n-1=\backslash left(2^n+1\backslash right)\backslash left(2^n-1\backslash right)$"

This proof looks wrong to me. You can't express a base 4 repunit as $4^n\; -\; 1$. ($5\; \backslash ne\; 4^1\; -\; 1$ and $5\; \backslash ne\; 4^2\; -\; 1\; =\; 15$ Is the proof wrong or am I losing my mind? — Preceding unsigned comment added by 128.237.217.71 (talk) 06:15, 5 December 2015 (UTC)

:You can express every base 4 repunit as $\backslash sum\_\{i=0\}^\{n-1\}\; 4^i=\backslash frac\{4^n-1\}\{4-1\}=\backslash frac\{\backslash left(2^n+1\backslash right)\backslash left(2^n-1\backslash right)\}\{3\}$ which explains the proof. Maybe one should add the sum for clarification though? I just struggled with the same thing. 130.180.121.126 (talk) 22:48, 9 March 2018 (UTC)

:I just saw that the same is true for the subsections on bases 8 and 9. 130.180.121.126 (talk) 22:54, 9 March 2018 (UTC)

This is original research on my part, but perhaps it should be added if we can find a reliable source.

Theorem: If the base b is u^v, then any repunit in base b has at most v digits.

Proof:

In general,

:$R\_m(n)\; =\; \backslash frac\; \{n^m-1\}\{n-1\}.$

If $b\; =\; u^v,$

:$R\_w(b)\; =\; \backslash frac\; \{\; u^\{vw\}\; -\; 1\}\; \{u^v-1\}\; =\; \backslash frac\; \{\backslash left(\; \{u^w-1\}\; \backslash right)\; R\_v\backslash left(u^w\backslash right)\}\; \{u^v-1\}$

If w>v, both components of the numerator are larger than the denominator, so the expression is composite.

A more careful proof can probably show that if w > 1, v and w cannot be relatively prime.

... or not. A counter example is reported above. — Arthur Rubin (talk) 22:47, 25 August 2018 (UTC)

It is stated in the article that if b is a power, then there is at most one repdigit in base b. No reference is given.... — Arthur Rubin (talk) 23:55, 25 August 2018 (UTC)

I don't know how much detail we need to go into. I cannot come up with a proof much simpler than:

:$R\_n(9)\; =\; \backslash frac\; \{\backslash left(3^n+1\backslash right)\; \backslash left(3^n+1\backslash right)\}\; 8\; .$

If n is greater than 1, 3ⁿ + 1 and 3ⁿ − 1 are even and greater than 4; hence, writing

$\backslash frac\; \{\backslash left(3^n+1\backslash right)\backslash left(3^n-1\backslash right)\}\; 8\; =\; \backslash frac\; \{3^n+1\}\{2^a\}\; \backslash frac\{3^n-1\}\{2^b\},$, with (a, b) = (1, 2) or (2, 1), both factors are greater than 1. — Arthur Rubin (talk) 07:35, 14 November 2018 (UTC)

What repunit is 2417 and 557? I think that you need an infinity calculator...

--190.245.110.53 (talk) 22:08, 12 August 2019 (UTC)

{{edit semi-protected|Repunit|answered=yes}}

Batalov and Propper found a new probable prime decimal repunit on 4/20/2021. Being one of the authors, I would like to add a sourced sentence about this new finding. This new probable prime also happens to be the largest currently known probable prime in the world.

Suggested edit:

now:

As of November 2012, all further candidates up to R2500000 have been tested, but no new probable primes have been found so far.

edit:

On April 20, 2021, Batalov and Propper found a new probable prime decimal repunit, R5794777. As of April 2021, all candidates up to R4300000 have been tested, and the search continues. Serge Batalov (talk) 05:38, 22 April 2021 (UTC)

:File:Red information icon with gradient background.svg Not done: please provide reliable sources that support the change you want to be made. ScottishFinnishRadish (talk) 10:59, 22 April 2021 (UTC)

::File:Yes check.svg Done The authoritative source {{oeis|004023}} was updated with both R5794777 and now R8177207. The article has been edited accordingly. -- P.T. Aufrette (talk) 01:33, 10 May 2021 (UTC)

{{Edit semi-protected|Repunit|answered=yes}}

To complete the allocation of prime factors (lower than 100) for decimal repunit numbers:

R33 = 67, R35 = 71, R41 = 83, R44 = 89, R46 = 47, R58 = 59, R60 = 61, R96 = 97. AcerSpes (talk) 11:06, 29 December 2022 (UTC)

:File:Red information icon with gradient background.svg Not done: Wikipedia is not a WP:INDISCRIMINATE collection of information. The summary is enough for reading on the information and people who'd like to check more can click on the external link. I see no reason to add this. Aaron Liu (talk) 15:21, 11 January 2023 (UTC)