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In [[recreational mathematics]], a '''repunit''' is a [[number]] like 11, 111, or 1111 that contains only the digit 1 &mdash; a more specific type of [[repdigit]]. The term stands for '''rep'''eated '''unit''' and was coined in 1966 by [[Albert H. Beiler]] in his book ''Recreations in the Theory of Numbers''.<ref>{{cite book|last1=Beiler|first1=Albert|title=Recreations in the Theory of Numbers: The Queen of Mathematics Entertains|date=2013|publisher=Dover Publications|location=New York|isbn=978-0-486-21096-4|page=83|edition=2|accessdate=21 June 2014}}</ref><!--- Original publication was 1964; did he coin repunit in that edition? or was this added in 1966? --->
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


A '''repunit prime''' is a repunit that is also a [[prime number]]. Primes that are repunits in [[Binary number|base 2]] are [[Mersenne prime]]s.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


== Definition ==
Registered users will be able to choose between the following three rendering modes:


The base-''b'' repunits are defined as (this ''b'' can be either positive or negative)
'''MathML'''
:<math>R_n^{(b)}={b^n-1\over{b-1}}\qquad\mbox{for }|b|\ge2, n\ge1.</math>
:<math forcemathmode="mathml">E=mc^2</math>
Thus, the number ''R''<sub>''n''</sub><sup>''(b)''</sup> consists of ''n'' copies of the digit 1 in base b representation. The first two repunits base ''b'' for ''n''=1 and ''n''=2 are
:<math>R_1^{(b)}={b-1\over{b-1}}= 1 \qquad \text{and} \qquad R_2^{(b)}={b^2-1\over{b-1}}= b+1\qquad\text{for}\ |b|\ge2.</math>


In particular, the ''[[decimal]] (base-10) repunits'' that are often referred to as simply ''repunits'' are defined as
<!--'''PNG''' (currently default in production)
:<math>R_n=R_n^{(10)}={10^n-1\over{10-1}}={10^n-1\over9}\qquad\mbox{for }n\ge1.</math>
:<math forcemathmode="png">E=mc^2</math>
Thus, the number ''R''<sub>''n''</sub> = ''R''<sub>''n''</sub><sup>''(10)''</sup> consists of ''n'' copies of the digit 1 in base 10 representation. The sequence of repunits base 10 starts with
: [[1 (number)|1]], [[11 (number)|11]], [[111 (number)|111]], 1111, 11111, 111111, ... {{OEIS|A002275}}.


Similarly, the repunits [[base 2]] are defined as
'''source'''
:<math>R_n^{(2)}={2^n-1\over{2-1}}={2^n-1}\qquad\mbox{for }n\ge1.</math>
:<math forcemathmode="source">E=mc^2</math> -->
Thus, the number ''R''<sub>''n''</sub><sup>''(2)''</sup> consists of ''n'' copies of the digit 1 in base 2 representation. In fact, the base-2 repunits are the well-known [[Mersenne prime|Mersenne number]]s ''M''<sub>''n''</sub>&nbsp;=&nbsp;2<sup>''n''</sup>&nbsp;&minus;&nbsp;1, they start with
:1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, ... {{OEIS|id=A000225}}


== Properties ==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
* Any repunit in any base having a composite number of digits is necessarily composite.  Only repunits (in any base) having a prime number of digits might be prime. This is a necessary but not sufficient condition.  For example,
*: ''R''<sub>35</sub><sup>(''b'')</sup> = {{repeat|1|35}} = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001,
:since 35 = 7 × 5 = 5 × 7.  This repunit factorization does not depend on the base ''b'' in which the repunit is expressed.
* Any positive multiple of the repunit ''R''<sub>''n''</sub><sup>(''b'')</sup> contains at least ''n'' nonzero digits in base ''b''.
* The only known numbers that are repunits with at least 3 digits in more than one base simultaneously are 31 (111 in base 5, 11111 in base 2) and 8191 (111 in base 90, 1111111111111 in base 2). The [[Goormaghtigh conjecture]] says there are only these two cases.
* Using the [[pigeon-hole principle]] it can be easily shown that for each ''n'' and ''b'' such that ''n''  and ''b'' are [[relative prime|relatively prime]] there exists a repunit in base ''b'' that is a multiple of ''n''. To see this consider repunits ''R''<sub>1</sub><sup>(''b'')</sup>,...,''R''<sub>''n''</sub><sup>(''b'')</sup>. Assume none of the ''R''<sub>''k''</sub><sup>(''b'')</sup> is divisible by ''n''. Because there are ''n'' repunits but only ''n''-1 non-zero residues modulo ''n'' there exist two repunits ''R''<sub>''i''</sub><sup>(''b'')</sup> and ''R''<sub>''j''</sub><sup>(''b'')</sup> with 1≤''i''<''j''≤''n'' such that ''R''<sub>''i''</sub><sup>(''b'')</sup> and ''R''<sub>''j''</sub><sup>(''b'')</sup> have the same residue modulo ''n''. It follows that ''R''<sub>''j''</sub><sup>(''b'')</sup> - ''R''<sub>''i''</sub><sup>(''b'')</sup> has residue 0 modulo ''n'', i.e. is divisible by ''n''. ''R''<sub>''j''</sub><sup>(''b'')</sup> - ''R''<sub>''i''</sub><sup>(''b'')</sup> consists of ''j'' - ''i'' ones followed by ''i'' zeroes. Thus, ''R''<sub>''j''</sub><sup>(''b'')</sup> - ''R''<sub>''i''</sub><sup>(''b'')</sup> = ''R''<sub>''j''-''i''</sub><sup>(''b'')</sup>  x ''b''<sup>''i''</sup> . Since ''n'' divides the left-hand side it also divides the right-hand side and since ''n'' and ''b'' are relative prime ''n'' must divide ''R''<sub>''j''-''i''</sub><sup>(''b'')</sup> [[proof by contradiction|contradicting]] the original assumption.
* The [[Feit–Thompson conjecture]] is that ''R''<sub>''q''</sub><sup>(''p'')</sup> never divides ''R''<sub>''p''</sub><sup>(''q'')</sup> for two distinct primes ''p'' and ''q''.


== Factorization of decimal repunits ==
==Demos==


(Prime factors colored {{color|red|red}} means "new factors", the prime factor divides ''R''<sub>''n''</sub> but not divides ''R''<sub>''k''</sub> for all ''k'' < ''n'') {{OEIS|id=A102380}}
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


{|
|-
||
{|
|-
|''R''<sub>1</sub> =||1
|-
|''R''<sub>2</sub> =||{{color|red|11}}
|-
|''R''<sub>3</sub> =||{{color|red|3}} · {{color|red|37}}
|-
|''R''<sub>4</sub> =||11 · {{color|red|101}}
|-
|''R''<sub>5</sub> =||{{color|red|41}} · {{color|red|271}}
|-
|''R''<sub>6</sub> =||3 · {{color|red|7}} · 11 · {{color|red|13}} · 37
|-
|''R''<sub>7</sub> =||{{color|red|239}} · {{color|red|4649}}
|-
|''R''<sub>8</sub> =||11 · {{color|red|73}} · 101 · {{color|red|137}}
|-
|''R''<sub>9</sub> =||3<sup>2</sup> · 37 · {{color|red|333667}}
|-
|''R''<sub>10</sub> =||11 · 41 · 271 · {{color|red|9091}}
|}
||
{|
|-
|''R''<sub>11</sub> =||{{color|red|21649}} · {{color|red|513239}}
|-
|''R''<sub>12</sub> =||3 · 7 · 11 · 13 · 37 · 101 · {{color|red|9901}}
|-
|''R''<sub>13</sub> =||{{color|red|53}} · {{color|red|79}} · {{color|red|265371653}}
|-
|''R''<sub>14</sub> =||11 · 239 · 4649 · {{color|red|909091}}
|-
|''R''<sub>15</sub> =||3 · {{color|red|31}} · 37 · 41 · 271 · {{color|red|2906161}}
|-
|''R''<sub>16</sub> =||11 · {{color|red|17}} · 73 · 101 · 137 · {{color|red|5882353}}
|-
|''R''<sub>17</sub> =||{{color|red|2071723}} · {{color|red|5363222357}}
|-
|''R''<sub>18</sub> =||3<sup>2</sup> · 7 · 11 · 13 · {{color|red|19}} · 37 · {{color|red|52579}} · 333667
|-
|''R''<sub>19</sub> =||{{color|red|{{repeat|1|19}}}}
|-
|''R''<sub>20</sub> =||11 · 41 · 101 · 271 · {{color|red|3541}} · 9091 · {{color|red|27961}}
|}
||
{|
|-
|''R''<sub>21</sub> =||3 · 37 · {{color|red|43}} · 239 · {{color|red|1933}} · 4649 · {{color|red|10838689}}
|-
|''R''<sub>22</sub> =||11<sup>2</sup> · {{color|red|23}} · {{color|red|4093}} · {{color|red|8779}} · 21649 · 513239
|-
|''R''<sub>23</sub> =||{{color|red|{{repeat|1|23}}}}
|-
|''R''<sub>24</sub> =||3 · 7 · 11 · 13 · 37 · 73 · 101 · 137 · 9901 · {{color|red|99990001}}
|-
|''R''<sub>25</sub> =||41 · 271 · {{color|red|21401}} · {{color|red|25601}} · {{color|red|182521213001}}
|-
|''R''<sub>26</sub> =||11 · 53 · 79 · {{color|red|859}} · 265371653 · {{color|red|1058313049}}
|-
|''R''<sub>27</sub> =||3<sup>3</sup> · 37 · {{color|red|757}} · 333667 · {{color|red|440334654777631}}
|-
|''R''<sub>28</sub> =||11 · {{color|red|29}} · 101 · 239 · {{color|red|281}} · 4649 · 909091 · {{color|red|121499449}}
|-
|''R''<sub>29</sub> =||{{color|red|3191}} · {{color|red|16763}} · {{color|red|43037}} · {{color|red|62003}} · {{color|red|77843839397}}
|-
|''R''<sub>30</sub> =||3 · 7 · 11 · 13 · 31 · 37 · 41 · {{color|red|211}} · {{color|red|241}} · 271 · {{color|red|2161}} · 9091 · 2906161
|}
|}


Smallest prime factor of ''R''<sub>''n''</sub> are
* accessibility:
:1, 11, 3, 11, 41, 3, 239, 11, 3, 11, 21649, 3, 53, 11, 3, 11, 2071723, 3, 1111111111111111111, 11, 3, 11, 11111111111111111111111, 3, 41, 11, 3, 11, 3191, 3, 2791, 11, 3, 11, 41, 3, 2028119, 11, 3, 11, 83, 3, 173, 11, 3, 11, 35121409, 3, 239, 11, ... {{OEIS|id=A067063}}
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


==Repunit primes==
==Test pages ==
The definition of repunits was motivated by recreational mathematicians looking for [[integer factorization|prime factors]] of such numbers.


It is easy to show that if ''n'' is divisible by ''a'', then ''R''<sub>''n''</sub><sup>(''b'')</sup> is divisible by ''R''<sub>''a''</sub><sup>(''b'')</sup>:
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


:<math>R_n^{(b)}=\frac{1}{b-1}\prod_{d|n}\Phi_d(b)</math>
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
where <math>\Phi_d(x)</math> is the <math>d^\mathrm{th}</math> [[cyclotomic polynomial]] and ''d'' ranges over the divisors of ''n''. For ''p'' prime, <math>\Phi_p(x)=\sum_{i=0}^{p-1}x^i</math>, which has the expected form of a repunit when ''x'' is substituted with ''b''.
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
For example, 9 is divisible by 3, and thus ''R''<sub>9</sub> is divisible by ''R''<sub>3</sub>&mdash;in fact, 111111111&nbsp;=&nbsp;111&nbsp;·&nbsp;1001001. The corresponding cyclotomic polynomials <math>\Phi_3(x)</math> and <math>\Phi_9(x)</math> are <math>x^2+x+1</math> and <math>x^6+x^3+1</math> respectively. Thus, for ''R''<sub>''n''</sub> to be prime ''n'' must necessarily be prime.
But it is not sufficient for ''n'' to be prime; for example, ''R''<sub>3</sub>&nbsp;=&nbsp;111&nbsp;=&nbsp;3&nbsp;·&nbsp;37 is not prime. Except for this case of ''R''<sub>3</sub>,  ''p'' can only divide ''R''<sub>''n''</sub> for prime ''n'' if ''p = 2kn + 1'' for some ''k''.
 
=== Decimal repunit primes ===
''R''<sub>''n''</sub> is prime for ''n''&nbsp;=&nbsp;2,&nbsp;19,&nbsp;23, 317, 1031, ... (sequence [[OEIS:A004023|A004023]] in [[OEIS]]). ''R''<sub>49081</sub> and ''R''<sub>86453</sub> are [[probable prime|probably prime]].  On April 3, 2007 [[Harvey Dubner]] (who also found ''R''<sub>49081</sub>) announced that ''R''<sub>109297</sub> is a probable prime.<ref>Harvey Dubner, ''[http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0704&L=nmbrthry&T=0&P=178 New Repunit R(109297)]''</ref> He later announced there are no others from ''R''<sub>86453</sub> to ''R''<sub>200000</sub>.<ref>Harvey Dubner, ''[http://tech.groups.yahoo.com/group/primeform/message/8546 Repunit search limit]''</ref> On July 15, 2007 Maksym Voznyy announced ''R''<sub>270343</sub> to be probably prime,<ref>Maksym Voznyy, ''[https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0707&L=NMBRTHRY&P=5903 New PRP Repunit R(270343)]''</ref> along with his intent to search to 400000. As of November 2012, all further candidates up to ''R''<sub>2500000</sub> have been tested, but no new probable primes have been found so far.
 
It has been conjectured that there are infinitely many repunit primes<ref>Chris Caldwell, "[http://primes.utm.edu/glossary/page.php?sort=Repunit The Prime Glossary: repunit]" at The [[Prime Pages]].</ref> and they seem to occur roughly as often as the [[prime number theorem]] would predict: the exponent of the ''N''th repunit prime is generally around a fixed multiple of the exponent of the (''N''-1)th.
 
The prime repunits are a trivial subset of the [[permutable prime]]s, i.e., primes that remain prime after any [[permutation]] of their digits.
 
=== Base 2 repunit primes ===
{{main|Mersenne prime}}
 
Base 2 repunit primes are called [[Mersenne prime]]s.
 
=== Base 3 repunit primes ===
The first few base 3 repunit primes are
: 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 {{OEIS|A076481}}
corresponding to <math>n</math> of
: 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, ... {{OEIS|A028491}}.
 
=== Base 4 repunit primes ===
The only base 4 repunit prime is 5 (<math>11_4</math>). <math>4^n-1=\left(2^n+1\right)\left(2^n-1\right)</math>, and 3 always divides <math>2^n+1</math> when ''n'' is odd and <math>2^n-1</math> when ''n'' is even. For ''n'' greater than 2, both <math>2^n+1</math> and <math>2^n-1</math> are greater than 3, so removing the factor of 3 still leaves two factors greater than 1, so the number cannot be prime.
 
===Base 5 repunit primes===
The first few base 5 repunit primes are
: 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531, 35032461608120426773093239582247903282006548546912894293926707097244777067146515037165954709053039550781, 815663058499815565838786763657068444462645532258620818469829556933715405574685778402862015856733535201783524826169013977050781 {{OEIS|A086122}}
corresponding to <math>n</math> of
: 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, ... {{OEIS|A004061}}.
 
===Base 6 repunit primes===
The first few base 6 repunit primes are
: 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371, 133733063818254349335501779590081460423013416258060407531857720755181857441961908284738707408499507 {{OEIS|A165210}}
corresponding to <math>n</math> of
: 2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, ... {{OEIS|A004062}}
 
===Base 7 repunit primes===
The first few base 7 repunit primes are
: 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457,<br/>138502212710103408700774381033135503926663324993317631729227790657325163310341833227775945426052637092067324133850503035623601
corresponding to <math>n</math> of
: 5, 13, 131, 149, 1699, ... {{OEIS|A004063}}
 
===Base 8 and 9 repunit primes===
The only base 8 ''or'' base 9 repunit prime is [[73 (number)|73]] (<math>111_8</math>).  <math>8^n-1=\left(4^n+2^n+1\right)\left(2^n-1\right)</math>, and 7 divides <math>4^n+2^n+1</math> when ''n'' is not divisible by 3 and <math>2^n-1</math> when ''n'' is a multiple of 3. <math>9^n-1=\left(3^n+1\right)\left(3^n-1\right)</math>, and 2 always divides both <math>3^n+1</math> and <math>3^n-1</math>.
 
===Base 12 repunit primes===
The first few base 12 repunit primes are
: 13, 157, 22621, 29043636306420266077, 435700623537534460534556100566709740005056966111842089407838902783209959981593077811330507328327968191581, 388475052482842970801320278964160171426121951256610654799120070705613530182445862582590623785872890159937874339918941
corresponding to <math>n</math> of
: 2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, ... {{OEIS|A004064}}
 
===Base 20 repunit primes===
The first few base 20 repunit primes are
: 421, 10778947368421, 689852631578947368421
corresponding to <math>n</math> of
: 3, 11, 17, 1487, ... {{OEIS|A127995}}
 
===The smallest repunit prime (''p''>2) of any natural number base ''b''===
 
The list is about all bases up to 300. {{OEIS|id=A128164}}
 
{| class="wikitable"
|'''Base'''
|'''+1'''
|'''+2'''
|'''+3'''
|'''+4'''
|'''+5'''
|'''+6'''
|'''+7'''
|'''+8'''
|'''+9'''
|'''+10'''
|'''+11'''
|'''+12'''
|'''+13'''
|'''+14'''
|'''+15'''
|'''+16'''
|'''+17'''
|'''+18'''
|'''+19'''
|'''+20'''
|-
|'''0+'''
|3
|3
|3
|None
|3
|3
|5
|3
|None
|19
|17
|3
|5
|3
|3
|None
|3
|25667
|19
|3
|-
|'''20+'''
|3
|5
|5
|3
|None
|7
|3
|5
|5
|5
|7
|None
|3
|13
|313
|None
|13
|3
|349
|5
|-
|'''40+'''
|3
|1319
|5
|5
|19
|7
|127
|19
|None
|3
|4229
|103
|11
|3
|17
|7
|3
|41
|3
|7
|-
|'''60+'''
|7
|3
|5
|None
|19
|3
|19
|5
|3
|29
|3
|7
|5
|5
|3
|41
|3
|3
|5
|3
|-
|'''80+'''
|None
|23
|5
|17
|5
|11
|7
|61
|3
|3
|4421
|439
|7
|5
|7
|3343
|17
|13
|3
|None
|-
|'''100+'''
|3
|59
|19
|97
|3
|149
|17
|449
|17
|3
|3
|79
|23
|29
|7
|59
|3
|5
|3
|5
|-
|'''120+'''
|None
|5
|43
|599
|None
|7
|5
|7
|5
|37
|3
|47
|13
|5
|1171
|227
|11
|3
|163
|79
|-
|'''140+'''
|3
|1231
|3
|None
|5
|7
|3
|1201
|7
|3
|13
|&#62;10000
|3
|5
|3
|7
|17
|7
|13
|7
|-
|'''160+'''
|3
|3
|7
|3
|5
|137
|3
|3
|None
|17
|181
|5
|3
|3251
|5
|3
|5
|347
|19
|7
|-
|'''180+'''
|17
|167
|223
|&#62;10000
|&#62;10000
|7
|37
|3
|3
|13
|17
|3
|5
|3
|11
|None
|31
|5
|577
|&#62;10000
|-
|'''200+'''
|271
|37
|3
|5
|19
|3
|13
|5
|3
|&#62;10000
|41
|11
|137
|191
|3
|None
|281
|3
|13
|7
|-
|'''220+'''
|7
|5
|239
|11
|None
|127
|5
|461
|11
|5333
|3
|953
|113
|61
|7
|3
|7
|7
|5
|109
|-
|'''240+'''
|17
|19
|None
|3331
|3
|3
|17
|41
|5
|127
|7
|541
|19
|5
|5
|None
|23
|11
|2011
|5
|-
|'''260+'''
|31
|197
|5
|7
|5
|3
|13
|11
|&#62;10000
|241
|41
|3
|37
|5
|5
|31
|5
|3
|3
|7
|-
|'''280+'''
|&#62;10000
|7
|29
|2473
|5
|13
|3
|3
|None
|3
|13
|5
|3
|7
|17
|41
|17
|53
|113
|7
|}
 
There are only [[probable prime]]s for that ''b'' = 18, 51, 91, 96, 174, 230, 244, 259, and 284.
 
No known repunit primes or PRPs for that ''b'' = 152, 184, 185, 200, 210, 269, and 281.
 
Because of the algebra factorization, there are no repunit primes for that ''b'' = 4, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 125, 144, 169, 196, 216, 225, 243, 256, and 289. {{OEIS|id=A126589}}
 
It is expected that all odd primes are in the list.
 
For [[negative base]]s (up to −300), see [[Wagstaff prime#Generalizations|Wagstaff prime]].
 
===The smallest natural number base ''b'' that <math>R_p</math> is prime for prime ''p''===
 
The list is about the first 100 primes. {{OEIS|id=A066180}}
 
{|class="wikitable"
|''p''
|2
|3
|5
|7
|11
|13
|17
|19
|23
|29
|31
|37
|41
|43
|47
|53
|59
|61
|67
|71
|-
|min ''b''
|2
|2
|2
|2
|5
|2
|2
|2
|10
|6
|2
|61
|14
|15
|5
|24
|19
|2
|46
|3
|-
|''p''
|73
|79
|83
|89
|97
|101
|103
|107
|109
|113
|127
|131
|137
|139
|149
|151
|157
|163
|167
|173
|-
|min ''b''
|11
|22
|41
|2
|12
|22
|3
|2
|12
|86
|2
|7
|13
|11
|5
|29
|56
|30
|44
|60
|-
|''p''
|179
|181
|191
|193
|197
|199
|211
|223
|227
|229
|233
|239
|241
|251
|257
|263
|269
|271
|277
|281
|-
|min ''b''
|304
|5
|74
|118
|33
|156
|46
|183
|72
|606
|602
|223
|115
|37
|52
|104
|41
|6
|338
|217
|-
|''p''
|283
|293
|307
|311
|313
|317
|331
|337
|347
|349
|353
|359
|367
|373
|379
|383
|389
|397
|401
|409
|-
|min ''b''
|13
|136
|220
|162
|35
|10
|218
|19
|26
|39
|12
|22
|67
|120
|195
|48
|54
|463
|38
|41
|-
|''p''
|419
|421
|431
|433
|439
|443
|449
|457
|461
|463
|467
|479
|487
|491
|499
|503
|509
|521
|523
|541
|-
|min ''b''
|17
|808
|404
|46
|76
|793
|38
|28
|215
|37
|236
|59
|15
|514
|260
|498
|6
|2
|95
|3
|}
 
The values of ''b'' that are [[perfect power]]s do not appear in this list, because they cannot be the base of a generalized repunit prime.<ref>{{citation
| last = Dubner | first = Harvey
| doi = 10.2307/2153263
| issue = 204
| journal = Mathematics of Computation
| mr = 1185243
| pages = 927–930
| title = Generalized repunit primes
| volume = 61
| year = 1993}}.</ref>
 
===List of repunit primes base ''b''===
{|class="wikitable"
|''b''
|''n'' which <math>R_n(b)</math> is prime (some large terms are only PRP, these ''n'''s are checked up to 100000)
|OEIS sequence
|-
|−30
|2, 139, 173, 547, 829, 2087, 2719, 3109, 10159, 56543, 80599, ...
|{{OEIS link|id=A071382}}
|-
|−29
|7, ...
|
|-
|−28
|3, 19, 373, 419, 491, 1031, 83497, ...
|{{OEIS link|id=A071381}}
|-
|−27
|None (Algebra)
|
|-
|−26
|11, 109, 227, 277, 347, 857, 2297, 9043, ...
|{{OEIS link|id=A071380}}
|-
|−25
|3, 7, 23, 29, 59, 1249, 1709, 1823, 1931, 3433, 8863, 43201, 78707, ...
|{{OEIS link|id=A057191}}
|-
|−24
|2, 7, 11, 19, 2207, 2477, 4951, ...
|{{OEIS link|id=A057190}}
|-
|−23
|11, 13, 67, 109, 331, 587, 24071, 29881, 44053, ...
|{{OEIS link|id=A057189}}
|-
|−22
|3, 5, 13, 43, 79, 101, 107, 227, 353, 7393, 50287, ...
|{{OEIS link|id=A057188}}
|-
|−21
|3, 5, 7, 13, 37, 347, 17597, 59183, 80761, 210599, ...
|{{OEIS link|id=A057187}}
|-
|−20
|2, 5, 79, 89, 709, 797, 1163, 6971, 140053, 177967, ...
|{{OEIS link|id=A057186}}
|-
|−19
|17, 37, 157, 163, 631, 7351, 26183, 30713, 41201, 77951, ...
|{{OEIS link|id=A057185}}
|-
|−18
|2, 3, 7, 23, 73, 733, 941, 1097, 1933, 4651, 481147, ...
|{{OEIS link|id=A057184}}
|-
|−17
|7, 17, 23, 47, 967, 6653, 8297, 41221, 113621, 233689, 348259, ...
|{{OEIS link|id=A057183}}
|-
|−16
|3, 5, 7, 23, 37, 89, 149, 173, 251, 307, 317, 30197, 1025393, ...
|{{OEIS link|id=A057182}}
|-
|−15
|3, 7, 29, 1091, 2423, 54449, 67489, 551927, ...
|{{OEIS link|id=A057181}}
|-
|−14
|2, 7, 53, 503, 1229, 22637, 1091401, ...
|{{OEIS link|id=A057180}}
|-
|−13
|3, 11, 17, 19, 919, 1151, 2791, 9323, 56333, ...
|{{OEIS link|id=A057179}}
|-
|−12
|2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ...
|{{OEIS link|id=A057178}}
|-
|−11
|5, 7, 179, 229, 439, 557, 6113, 223999, 327001, ...
|{{OEIS link|id=A057177}}
|-
|−10
|5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ...
|{{OEIS link|id=A001562}}
|-
|−9
|3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, ...
|{{OEIS link|id=A057175}}
|-
|−8
|2 (Algebra)
|
|-
|−7
|3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, ...
|{{OEIS link|id=A057173}}
|-
|−6
|2, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, ...
|{{OEIS link|id=A057172}}
|-
|−5
|5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, ...
|{{OEIS link|id=A057171}}
|-
|−4
|2, 3 ([[Aurifeuillean factorization]])
|
|-
|−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, ...
|{{OEIS link|id=A007658}}
|-
|−2
|3, 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, ... (1 is not prime)
|{{OEIS link|id=A000978}}
|-
|−1
|None (1 is not prime)
|
|-
|0
|None (1 is not prime)
|
|-
|1
|2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, ... (All primes)
|{{OEIS link|id=A000040}}
|-
|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, ...
|{{OEIS link|id=A000043}}
|-
|3
|3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, ...
|{{OEIS link|id=A028491}}
|-
|4
|2 (Algebra)
|
|-
|5
|3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, ...
|{{OEIS link|id=A004061}}
|-
|6
|2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, ...
|{{OEIS link|id=A004062}}
|-
|7
|5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ...
|{{OEIS link|id=A004063}}
|-
|8
|3 (Algebra)
|
|-
|9
|None (Algebra)
|
|-
|10
|2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ...
|{{OEIS link|id=A004023}}
|-
|11
|17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, ...
|{{OEIS link|id=A005808}}
|-
|12
|2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, ...
|{{OEIS link|id=A004064}}
|-
|13
|5, 7, 137, 283, 883, 991, 1021, 1193, 3671, 18743, 31751, 101089, ...
|{{OEIS link|id=A016054}}
|-
|14
|3, 7, 19, 31, 41, 2687, 19697, 59693, 67421, ...
|{{OEIS link|id=A006032}}
|-
|15
|3, 43, 73, 487, 2579, 8741, 37441, 89009, ...
|{{OEIS link|id=A006033}}
|-
|16
|2 (Algebra)
|
|-
|17
|3, 5, 7, 11, 47, 71, 419, 4799, 35149, 54919, 74509, ...
|{{OEIS link|id=A006034}}
|-
|18
|2, 25667, 28807, 142031, 157051, 180181, ...
|{{OEIS link|id=A133857}}
|-
|19
|19, 31, 47, 59, 61, 107, 337, 1061, 9511, 22051, 209359, ...
|{{OEIS link|id=A006035}}
|-
|20
|3, 11, 17, 1487, 31013, 48859, 61403, ...
|{{OEIS link|id=A127995}}
|-
|21
|3, 11, 17, 43, 271, 156217, 328129, ...
|{{OEIS link|id=A127996}}
|-
|22
|2, 5, 79, 101, 359, 857, 4463, 9029, 27823, ...
|{{OEIS link|id=A127997}}
|-
|23
|5, 3181, 61441, 91943, ...
|{{OEIS link|id=A204940}}
|-
|24
|3, 5, 19, 53, 71, 653, 661, 10343, 49307, ...
|{{OEIS link|id=A127998}}
|-
|25
|None (Algebra)
|
|-
|26
|7, 43, 347, 12421, 12473, 26717, ...
|{{OEIS link|id=A127999}}
|-
|27
|3 (Algebra)
|
|-
|28
|2, 5, 17, 457, 1423, ...
|{{OEIS link|id=A128000}}
|-
|29
|5, 151, 3719, 49211, 77237, ...
|{{OEIS link|id=A181979}}
|-
|30
|2, 5, 11, 163, 569, 1789, 8447, 72871, 78857, 82883, ...
|{{OEIS link|id=A098438}}
|}
 
For more information, see [http://www.primenumbers.net/Henri/us/MersFermus.htm Repunit primes in base −50 to 50], [http://www.fermatquotient.com/PrimSerien/GenRepu Repunit primes in base 2 to 150], [http://www.fermatquotient.com/PrimSerien/GenRepuP Repunit primes in base −150 to −2], and [https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf Repunit primes in base −200 to −2].
 
=== Algebra factorization of repunit numbers ===
If ''b'' is a [[perfect power]] (can be written as ''m''<sup>''n''</sup>, with ''m'', ''n'' integers, ''n'' > 1) differs from 1, then there is at most one repunit in base ''b''. If ''n'' is a [[prime power]] (can be written as ''p''<sup>''r''</sup>, with ''p'' prime, ''r'' integer, ''p'', ''r'' >0), then all repunit in base ''b'' are not prime aside from ''R<sub>p</sub>'' and ''R<sub>2</sub>''. ''R<sub>p</sub>'' can be either prime or composite, the former examples, ''b'' = -216, -128, 4, 8, 16, 27, 36, 100, 128, 256, etc., the letter examples, ''b'' = -243, -125, -64, -32, -27, -8, 9, 25, 32, 49, 81, 121, 125, 144, 169, 196, 216, 225, 243, 289, etc., and ''R<sub>2</sub>'' can be prime (when ''p'' differs from 2) only if ''b'' is negative, a power of -2, for example, ''b'' = -8, -32, -128, -8192, etc., in fact, the ''R<sub>2</sub>'' can also be composite, for example, ''b'' = -512, -2048, -32768, etc. If ''n'' is not a prime power, then no base ''b'' repunit prime exists, for example, ''b'' = 64, 729 (with ''n'' = 6), ''b'' = 1024 (with ''n'' = 10), and ''b'' = -1 or 0 (with ''n'' any natural number). Another special situation is ''b'' = -4''k''<sup>4</sup>, with ''k'' positive integer, which has the [[aurifeuillean factorization]], for example, ''b'' = -4 (with ''k'' = 1, then ''R<sub>2</sub>'' and ''R<sub>3</sub>'' are primes), and ''b'' = -64, -324, -1024, -2500, -5184, ... (with ''k'' = 2, 3, 4, 5, 6, ..., then no base ''b'' repunit prime exists). It is also conjectured that when ''b'' is neither a perfect power nor -4''k''<sup>4</sup> with ''k'' positive integer, then there are infinity many base ''b'' repunit primes.
 
==History==
Although they were not then known by that name, repunits in base 10 were studied by many mathematicians during the nineteenth century in an effort to work out and predict the cyclic patterns of [[repeating decimal]]s.<ref>Dickson, Leonard Eugene and Cresse, G.H.; ''[[History of the Theory of Numbers]]''; pp. 164-167 ISBN 0-8218-1934-8</ref>
 
It was found very early on that for any prime ''p'' greater than 5, the [[Repeating decimal|period]] of the decimal expansion of 1/''p'' is equal to the length of the smallest repunit number that is divisible by ''p''. Tables of the period of reciprocal of primes up to 60,000 had been published by 1860 and permitted the [[integer factorization|factorization]] by such mathematicians as Reuschle of all repunits up to ''R<sub>16</sub>'' and many larger ones. By 1880, even ''R<sub>17</sub>'' to ''R<sub>36</sub>'' had been factored<ref>Dickson and Cresse, pp. 164-167</ref> and it is curious that, though [[Édouard Lucas]] showed no prime below three million had period [[19 (number)|nineteen]], there was no attempt to test any repunit for primality until early in the twentieth century. The American mathematician [[Oscar Hoppe]] proved ''R<sub>19</sub>'' to be prime in 1916<ref>Francis, Richard L.; "Mathematical Haystacks: Another Look at Repunit Numbers" in The College Mathematics Journal, Vol. 19, No. 3. (May, 1988), pp. 240-246.</ref> and Lehmer and Kraitchik independently found ''R<sub>23</sub>'' to be prime in 1929.
 
Further advances in the study of repunits did not occur until the 1960s, when computers allowed many new factors of repunits to be found and the gaps in earlier tables of prime periods corrected. ''R<sub>317</sub>'' was found to be a [[probable prime]] circa 1966 and was proved prime eleven years later, when ''R<sub>1031</sub>'' was shown to be the only further possible prime repunit with fewer than ten thousand digits. It was proven prime in 1986, but searches for further prime repunits in the following decade consistently failed. However, there was a major side-development in the field of generalized repunits, which produced a large number of new primes and probable primes.
 
Since 1999, four further probably prime repunits have been found, but it is unlikely that any of them will be proven prime in the foreseeable future because of their huge size.
 
The [[Cunningham project]] endeavours to document the integer factorizations of (among other numbers) the repunits to base 2, 3, 5, 6, 7, 10, 11, and 12.
 
== Demlo numbers ==
The {{Anchor|Demlo number}} Demlo numbers<ref>{{MathWorld |title= Demlo Number |urlname=DemloNumber}}</ref> 1, 121, 12321, 1234321, ..., 12345678987654321, 1234567900987654321, 123456790120987654321, ..., {{OEIS|id=A002477}} were defined by [[D. R. Kaprekar]] as the squares of the repunits, resolving the uncertainty how to continue beyond the highest digit (9), and named after [[Demlo]] railway station 30 miles from Bombay on the then [[G.I.P. Railway]], where he thought of investigating them.
 
==See also==
 
* [[Repdigit]]
* [[Recurring decimal]]
* [[All one polynomial]] - Another generalization
* [[Goormaghtigh conjecture]]
* [[Wagstaff prime]] - can be thought of as repunit primes with [[negative base]] <math>b=-2</math>
 
==Notes==
{{reflist|2}}
 
==External links==
 
===Web sites===
*{{mathworld|urlname=Repunit|title=Repunit}}
*[http://www.cerias.purdue.edu/homes/ssw/cun/third/pmain901 The main tables] of the [http://www.cerias.purdue.edu/homes/ssw/cun/ Cunningham project].
*[http://primes.utm.edu/glossary/page.php?sort=Repunit Repunit] at [http://primes.utm.edu/ The Prime Pages] by [[Chris Caldwell]].
*[http://www.worldofnumbers.com/repunits.htm Repunits and their prime factors] at [http://www.worldofnumbers.com World!Of Numbers].
*[http://www.primes.viner-steward.org/andy/titans.html Prime generalized repunits] of at least 1000 decimal digits by Andy Steward
*[http://www.elektrosoft.it/matematica/repunit/repunit.htm Repunit Primes Project] Giovanni Di Maria's repunit primes page.
*[http://www.primenumbers.net/Henri/us/MersFermus.htm Generalized repunit primes in base -50 to 50]
 
===Books===
*S. Yates, ''Repunits and repetends''. ISBN 0-9608652-0-9.
*A. Beiler, ''Recreations in the theory of numbers''. ISBN 0-486-21096-0. Chapter 11.
*[[Paulo Ribenboim]], ''The New Book Of Prime Number Records''. ISBN 0-387-94457-5.
 
{{Classes of natural numbers}}
{{Prime number classes}}
 
[[Category:Integers]]
[[Category:Base-dependent integer sequences]]

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