Talk:Repunit

Repunits & repdigits

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

Finitely many repunit primes?

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)

p = 2kn + 1

"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.[1] PrimeHunter (talk) 00:09, 8 December 2007 (UTC)

Divisibility

"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)

Blowing My Own Trumpet

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)

Own article about repunit primes?

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)

Repunits in specific bases

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 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 [2] 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)

Invalid digits used in base-n

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)

Nonconstructive editing

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

$a\equiv b{\pmod {n}},$

then

$a^{N}\equiv b^{N}{\pmod {n}}.$

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)

Formatting "factorization of decimal repunits" tables

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 <br>s doesn't really solve the problem, and doesn't maintain proper alignment. — Arthur Rubin (talk) 18:13, 21 August 2014 (UTC)

My generalized repunit data

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)

Proof that 5 is the only base 4 repunit prime?

"The only base 4 repunit prime is 5 ($11_{4}$). $4^{n}-1=\left(2^{n}+1\right)\left(2^{n}-1\right)$" This proof looks wrong to me. You can't express a base 4 repunit as $4^{n}-1$. ($5\neq 4^{1}-1$ and $5\neq 4^{2}-1=15$ 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 $\sum _{i=0}^{n-1}4^{i}={\frac {4^{n}-1}{4-1}}={\frac {\left(2^{n}+1\right)\left(2^{n}-1\right)}{3}}$ 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)

Power bases

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,

$R_{m}(n)={\frac {n^{m}-1}{n-1}}.$

If $b=u^{v},$

$R_{w}(b)={\frac {u^{vw}-1}{u^{v}-1}}={\frac {\left({u^{w}-1}\right)R_{v}\left(u^{w}\right)}{u^{v}-1}}$

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)

Base 9 repunit primes

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

$R_{n}(9)={\frac {\left(3^{n}+1\right)\left(3^{n}+1\right)}{8}}.$

If n is greater than 1, 3ⁿ + 1 and 3ⁿ − 1 are even and greater than 4; hence, writing ${\frac {\left(3^{n}+1\right)\left(3^{n}-1\right)}{8}}={\frac {3^{n}+1}{2^{a}}}{\frac {3^{n}-1}{2^{b}}},$, with (a, b) = (1, 2) or (2, 1), both factors are greater than 1. — Arthur Rubin (talk) 07:35, 14 November 2018 (UTC)

Repunit of: 2417, 557

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)

Semi-protected edit request on 22 April 2021

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)

 Not done: please provide reliable sources that support the change you want to be made. ScottishFinnishRadish (talk) 10:59, 22 April 2021 (UTC)

 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)

Semi-protected edit request on 29 December 2022

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)

 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)