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| In [[recreational mathematics]], a '''repunit''' is a [[number]] like 11, 111, or 1111 that contains only the digit 1 — the simplest form of [[repdigit]]. The term stands for '''rep'''eated '''unit''' and was coined in 1966 by [[Albert H. Beiler]].
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| A '''repunit prime''' is a repunit that is also a [[prime number]]. In [[Binary number|binary]], these are the widely known [[Mersenne prime]]s.
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| == Definition ==
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| The base-''b'' repunits are defined as
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| :<math>R_n^{(b)}={b^n-1\over{b-1}}\qquad\mbox{for }b\ge2, n\ge1.</math>
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| 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
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| :<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>
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| In particular, the ''[[decimal]] (base-10) repunits'' that are often referred to as simply ''repunits'' are defined as
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| :<math>R_n=R_n^{(10)}={10^n-1\over{10-1}}={10^n-1\over9}\qquad\mbox{for }n\ge1.</math>
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| 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
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| : [[1 (number)|1]], [[11 (number)|11]], [[111 (number)|111]], 1111, ... {{OEIS|A002275}}.
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| Similarly, the repunits base 2 are defined as
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| :<math>R_n^{(2)}={2^n-1\over{2-1}}={2^n-1}\qquad\mbox{for }n\ge1.</math>
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| 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> = 2<sup>''n''</sup> − 1.
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| == Properties ==
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| * 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,
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| *: ''R''<sub>35</sub><sup>(''b'')</sup> = 11111111111111111111111111111111111 = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001,
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| :since 35 = 7 × 5 = 5 × 7. This repunit factorization does not depend on the base ''b'' in which the repunit is expressed.
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| * Any positive multiple of the repunit ''R''<sub>''n''</sub><sup>(''b'')</sup> contains at least ''n'' nonzero digits in base ''b''.
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| * 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.
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| * Using the [[pigeon-hole principle]] it can be easily shown that for each ''n'' and ''b'' such that ''n'' and ''b'' are [[relative 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 10<sup>''i''</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.
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| * 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''.
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| == Factorization of decimal repunits ==
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| {|
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| |-
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| {|
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| |-
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| |R<sub>1</sub> =||1
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| |-
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| |R<sub>2</sub> =||11
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| |-
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| |R<sub>3</sub> =||3 · 37
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| |-
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| |R<sub>4</sub> =||11 · 101
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| |-
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| |R<sub>5</sub> =||41 · 271
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| |-
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| |R<sub>6</sub> =||3 · 7 · 11 · 13 · 37
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| |-
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| |R<sub>7</sub> =||239 · 4649
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| |-
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| |R<sub>8</sub> =||11 · 73 · 101 · 137
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| |-
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| |R<sub>9</sub> =||3 · 3 · 37 · 333667
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| |-
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| |R<sub>10</sub> =||11 · 41 · 271 · 9091
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| |}
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| {|
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| |-
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| |R<sub>11</sub> =||21649 · 513239
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| |-
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| |R<sub>12</sub> =||3 · 7 · 11 · 13 · 37 · 101 · 9901
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| |-
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| |R<sub>13</sub> =||53 · 79 · 265371653
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| |-
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| |R<sub>14</sub> =||11 · 239 · 4649 · 909091
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| |-
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| |R<sub>15</sub> =||3 · 31 · 37 · 41 · 271 · 2906161
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| |-
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| |R<sub>16</sub> =||11 · 17 · 73 · 101 · 137 · 5882353
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| |-
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| |R<sub>17</sub> =||2071723 · 5363222357
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| |R<sub>18</sub> =||3 · 3 · 7 · 11 · 13 · 19 · 37 · 52579 · 333667
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| |R<sub>19</sub> =||1111111111111111111
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| |-
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| |R<sub>20</sub> =||11 · 41 · 101 · 271 · 3541 · 9091 · 27961
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| |}
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| |}
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| ==Repunit primes==
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| The definition of repunits was motivated by recreational mathematicians looking for [[integer factorization|prime factors]] of such numbers.
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| 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>:
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| :<math>R_n^{(b)}=\frac{1}{b-1}\prod_{d|n}\Phi_d(b)</math>
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| 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''.
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| For example, 9 is divisible by 3, and thus ''R''<sub>9</sub> is divisible by ''R''<sub>3</sub>—in fact, 111111111 = 111 · 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.
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| But it is not sufficient for ''n'' to be prime; for example, ''R''<sub>3</sub> = 111 = 3 · 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''.
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| === Decimal repunit primes ===
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| ''R''<sub>''n''</sub> is prime for ''n'' = 2, 19, 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, ''[http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0707&L=nmbrthry&T=0&P=1086 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.
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| 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.
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| The prime repunits are a trivial subset of the [[permutable prime]]s, i.e., primes that remain prime after any [[permutation]] of their digits.
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| === Base-2 repunit primes ===
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| {{main|Mersenne prime}}
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| Base-2 repunit primes are called [[Mersenne prime]]s.
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| === Base-3 repunit primes ===
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| The first few base-3 repunit primes are
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| : 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013, ... {{OEIS|A076481}},
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| corresponding to <math>n</math> of
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| : 3, 7, 13, 71, 103, ... {{OEIS|A028491}}.
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| === Base-4 repunit primes ===
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| 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.
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| ===Base 5 repunit primes===
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| The first few base-5 ([[quinary]]) repunit primes are
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| : 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, {{OEIS|A086122}}
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| corresponding to <math>n</math> of
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| : 3, 7, 11, 13, 47, ... {{OEIS|A004061}}.
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| ===Base 6 repunit primes===
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| The first few base-6 repunit primes are
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| : 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371, ..., {{OEIS|A165210}}
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| corresponding to <math>n</math> of
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| : 2, 3, 7, 29, 71, ... {{OEIS|A004062}}
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| ===Base 7 repunit primes===
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| The first few base 7 repunit primes are
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| : 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457,<br/>138502212710103408700774381033135503926663324993317631729227790657325163310341833227775945426052637092067324133850503035623601
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| corresponding to <math>n</math> of
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| : 5, 13, 131, 149, ... {{OEIS|A004063}}
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| ===Base 8 and 9 repunit primes===
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| The only [[octal|base-8]] ''or'' [[nonary|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>.
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| ===Base 12 ([[duodecimal]]) repunit primes===
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| The first few base 12 repunit primes are
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| : 13, 157, 22621, 29043636306420266077, 435700623537534460534556100566709740005056966111842089407838902783209959981593077811330507328327968191581, 388475052482842970801320278964160171426121951256610654799120070705613530182445862582590623785872890159937874339918941
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| corresponding to <math>n</math> of
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| : 2, 3, 5, 19, 97, 109, 317, 353, 701, ... {{OEIS|A004064}}
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| ===Base 20 ([[vigesimal]]) repunit primes===
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| The only known vigesimal (base 20) repunit primes or [[probable prime]]s are for <math>n</math> of
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| : 3, 11, 17, 1487, 31013, 48859, 61403 {{OEIS|A127995}}
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| The first three of these in decimal are
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| : 421, 10778947368421 and 689852631578947368421
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| ==History==
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| 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 [[recurring 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>
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| 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> 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 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.
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| 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.
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| 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.
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| 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.
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| == Demlo numbers ==
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| The {{Anchor|Demlo number}} Demlo numbers<ref>{{MathWorld |title= Demlo Number |urlname=DemloNumber}}</ref> 1, 121, 12321, 1234321, … 12345678987654321, 1234567900987654321, 123456790120987654321, … 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.
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| ==See also==
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| * [[Repdigit]]
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| * [[Recurring decimal]]
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| * [[All one polynomial]] - Another generalization
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| * [[Goormaghtigh conjecture]]
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| ==Notes==
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| {{reflist|2}}
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| ==External links==
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| ===Web sites===
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| *{{mathworld|urlname=Repunit|title=Repunit}}
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| *[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].
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| *[http://primes.utm.edu/glossary/page.php?sort=Repunit Repunit] at [http://primes.utm.edu/ The Prime Pages] by [[Chris Caldwell]].
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| *[http://www.worldofnumbers.com/repunits.htm Repunits and their prime factors] at [http://www.worldofnumbers.com World!Of Numbers].
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| *[http://www.primes.viner-steward.org/andy/titans.html Prime generalized repunits] of at least 1000 decimal digits by Andy Steward
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| *[http://www.elektrosoft.it/matematica/repunit/repunit.htm Repunit Primes Project] Giovanni Di Maria's repunit primes page.
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| *[http://homepage2.nifty.com/m_kamada/math/11111.htm Factorizations of 11...11 (Repunit)] by Makoto Kamada
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| ===Books===
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| *S. Yates, ''Repunits and repetends''. ISBN 0-9608652-0-9.
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| *A. Beiler, ''Recreations in the theory of numbers''. ISBN 0-486-21096-0. Chapter 11, of course.
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| *[[Paulo Ribenboim]], ''The New Book Of Prime Number Records''. ISBN 0-387-94457-5.
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| {{Classes of natural numbers}}
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| [[Category:Integers]]
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| [[Category:Base-dependent integer sequences]]
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