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. &mdash; 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. 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  &lt; 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  &lt; 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  &lt; 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  &lt; 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 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  &lt; 04:08, 13 March 2011 (UTC)

The only guideline which might be applicable here seems to be Manual of Style (dates and 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  &lt; 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 Rn 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)

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 $$4^n - 1$$. ($$ 5 \ne 4^1 - 1 $$ and $$ 5 \ne 4^2 - 1 = 15 $$ 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 $$\sum_{i=0}^{n-1} 4^i=\frac{4^n-1}{4-1}=\frac{\left(2^n+1\right)\left(2^n-1\right)}{3}$$ 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 uv, 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 &minus; 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)
 * 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)
 * Yes check.svg Done The authoritative source 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)
 * 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)