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A '''repunit''' is a [[number]] like 11, 111, or 1111. It only has the [[digit]] 1 in it. It is 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''.{{refn|group=note|Albert H. Beiler coined the term “repunit number” as follows:<blockquote>A number which consists of a repeated of a single digit is sometimes called a monodigit number, and for convenience the author has used the term “repunit number” (repeated unit) to represent monodigit numbers consisting solely of the digit 1.<ref>{{Harvnb|Beiler|2013|pp=83}}</ref></blockquote>}}<!--- Original publication was 1964; did he coin repunit in that edition? or was this added in 1966? --->

A '''repunit prime''' is a repunit that is also a [[prime number]]. Primes that are repunits in [[Binary number|base-2]] are [[Mersenne prime|Mersenne primes]].

== Definition ==

The base-''b'' repunits can be written as this where ''b'' is the [[Base (mathematics)|base]] and ''n'' is the number that you are checking in whether or not it is a repunit:

: <math>R_n^{(b)}\equiv 1 + b + b^2 + \cdots + b^{n-1} = {b^n-1\over{b-1}}\qquad\mbox{for }|b|\ge2, n\ge1.</math>

This means that the number ''R''<sub>''n''</sub><sup>(''b'')</sup> is made of of of ''n'' copies of the digit 1 in base-''b'' representation. The first two repunits base-''b'' for ''n''&#x2009;=&#x2009;1 and ''n''&#x2009;=&#x2009;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>

The first of repunits in base-10 are with

: [[1 (number)|1]], [[11 (number)|11]], [[111 (number)|111]], 1111, 11111, 111111, ... {{OEIS|A002275}}.

Base-2 repunits are also [[Mersenne prime|Mersenne numbers]] ''M''<sub>''n''</sub>&nbsp;=&nbsp;2<sup>''n''</sup>&nbsp;&#x2212;&nbsp;1. They start with

: 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, ... {{OEIS|id=A000225}}.

== Factorization of decimal repunits ==

Prime factors that are {{color|red|red}} are "new factors" that haven't been mentioned before. Basically, the prime factor divides ''R''<sub>''n''</sub> but does not divide ''R''<sub>''k''</sub> for all ''k'' < ''n.'' {{OEIS|id=A102380}}<ref>For more information, see [http://stdkmd.net/nrr/repunit/ Factorization of repunit numbers].</ref>

{|
|
{|
|''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|{{loop|19|1}}}}
|-
|''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|{{loop|23|1}}}}
|-
|''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
|}
|}

The smallest [[Prime factorization|prime factors]] of ''R''<sub>''n''</sub> for ''n'' > 1 are

: 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}}

== Related pages ==
* [[Repeating decimal]]
* [[Repdigit]]

== Footnotes ==

=== Notes ===
{{reflist|group=note}}

=== References ===
{{reflist|30em}}

== Further reading ==

* {{Citation|last=Beiler|first=Albert H.|title=Recreations in the Theory of Numbers: The Queen of Mathematics Entertains|date=2013|url={{Google books|NbbbL9gMJ88C|Recreations in the theory of numbers|plainurl=yes}}|series=Dover Recreational Math|edition=2nd Revised|location=New York|publisher=Dover Publications|isbn=978-0-486-21096-4|author-link=Albert Beiler|orig-year=1964}}
* {{Citation|last1=Dickson|first1=Leonard Eugene|title=History of the Theory of Numbers|url={{Google books|XnwsAQAAIAAJ|History of the Theory of Numbers|plainurl=yes}}|year=1999|series=Volume I: Divisibility and primality|edition=2nd Reprinted|location=Providence, RI|publisher=AMS Chelsea Publishing|isbn=978-0-8218-1934-0|last2=Cresse|first2=G.H.|author-link=Leonard Eugene Dickson|author2-link=G. H. Cresse}}
* {{Citation|last=Francis|first=Richard L.|title=Mathematical Haystacks: Another Look at Repunit Numbers|journal=The College Mathematics Journal|volume=19|issue=3|pages=240–246|year=1988|doi=10.1080/07468342.1988.11973120|author-link=Richard L. Francis}}
* {{Citation|last1=Gunjikar|first1=K. R.|title=Theory of Demlo numbers|url=http://OEIS.org/A249605/a249605.pdf|journal=Journal of the University of Bombay|volume=VIII|issue=3|pages=3–9|year=1939|last2=Kaprekar|first2=D. R.|author-link=K. R. Gunjikar|author-link2=D. R. Kaprekar}}
* {{Citation|last=Kaprekar|first=D. R.|title=On Wonderful Demlo numbers|url=http://www.indianmathsociety.org.in/|journal=The Mathematics Student|volume=6|page=68|year=1938a|author-link=D. R. Kaprekar|access-date=2022-03-08|archive-date=2009-02-10|archive-url=https://web.archive.org/web/20090210075900/http://indianmathsociety.org.in/|url-status=dead}}
* {{Citation|last=Kaprekar|first=D. R.|title=Demlo numbers|journal=J. Phys. Sci. Univ. Bombay|volume=VII|issue=3|year=1938b|author-link=D. R. Kaprekar}}
* {{Citation|last=Kaprekar|first=D. R.|title=Demlo numbers|year=1948|place=Devlali, India|publisher=Khareswada|author-link=D. R. Kaprekar}}
* {{Citation|last=Ribenboim|first=Paulo|title=The New Book of Prime Number Records|date=1996-02-02|url={{Google books|2VTSBwAAQBAJ|The New Book of Prime Number Records|plainurl=yes}}|series=Computers and Medicine|edition=3rd|location=New York|publisher=Springer|isbn=978-0-387-94457-9|author-link=Paulo Ribenboim}}
* {{Citation|last=Yates|first=Samuel|title=Repunits and repetends|url={{Google books|3_vuAAAAMAAJ|Repunits and repetends|plainurl=yes}}|year=1982|location=FL|publisher=Delray Beach|isbn=978-0-9608652-0-8|author-link=Samuel Yates}}

== Other websites ==
* {{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 [[Prime Pages|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].
* [https://web.archive.org/web/20131019185910/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.
* [https://docs.google.com/document/d/e/2PACX-1vRha5-vd9Covl4k6I02lOEdWdhsXnBCeFT3FHyhMYTO1i7jtZdTV3wQw2yXgTNja5k_-XINgqL9VNmo/pub Smallest odd prime p such that (b^p-1)/(b-1) and (b^p+1)/(b+1) is prime for bases 2<=b<=1024]
* [http://stdkmd.net/nrr/repunit/ Factorization of repunit numbers]
* [http://www.primenumbers.net/Henri/us/MersFermus.htm Generalized repunit primes in base -50 to 50]

{{Classes of natural numbers}}

[[Category:Integer sequences]]