Editing Repunit (section)

==Repunit primes==
{{main list|List of repunit primes}}

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>:

:<math>R_n^{(b)}=\frac{1}{b-1}\prod_{d|n}\Phi_d(b),</math>

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''.

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'' = 2''kn'' + 1 for some ''k''.

=== Decimal repunit primes ===
''R''<sub>''n''</sub> is prime for ''n''&nbsp;=&nbsp;2,&nbsp;19,&nbsp;23, 317, 1031, 49081, 86453, 109297  ... (sequence [[OEIS:A004023|A004023]] in [[OEIS]]). 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> Serge Batalov and Ryan Propper found ''R''<sub>5794777</sub> and ''R''<sub>8177207</sub> to be probable primes on April 20 and May 8, 2021, respectively.<ref>{{cite OEIS|A004023|2=Indices of prime repunits: numbers n such that 11...111 (with n 1's) = (10^n - 1)/9 is prime.}}</ref> As of their discovery, each was the largest known probable prime. On March 22, 2022, probable prime ''R''<sub>49081</sub> was eventually proven to be a prime.<ref>{{cite web |url=https://primes.utm.edu/primes/page.php?id=133761 |title=PrimePage Primes: R(49081) |date=2022-03-21 |website=PrimePage Primes |access-date=2022-03-31}}</ref> On May 15, 2023, probable prime ''R''<sub>86453</sub> was eventually proven to be a prime.<ref>{{cite web |url=https://primes.utm.edu/primes/page.php?id=136044 |title=PrimePage Primes: R(86453) |date=2023-05-16 |website=PrimePage Primes |access-date=2023-05-16}}</ref> On May 26, 2025, probable prime ''R''<sub>109297</sub> was eventually proven to be a prime.<ref>{{cite web |url=https://primes.utm.edu/primes/page.php?id=140799 |title=PrimePage Primes: R(109297) |date=2025-05-27|website=PrimePage Primes |access-date=2025-05-27}}</ref>

It has been conjectured that there are infinitely many repunit primes<ref>{{cite web |author=Chris Caldwell |url=http://primes.utm.edu/glossary/page.php?sort=Repunit |work=The Prime Glossary |title=repunit |publisher=[[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.

Particular properties are

*The remainder of ''R''<sub>''n''</sub> modulo 3 is equal to the remainder of ''n'' modulo 3. Using 10<sup>''a''</sup> ≡ 1 (mod 3) for any ''a'' &ge; 0,<br />''n'' ≡ 0 (mod 3) ⇔ ''R''<sub>''n''</sub> ≡ 0 (mod 3) ⇔ ''R''<sub>''n''</sub> ≡ 0 (mod ''R''<sub>3</sub>),<br />''n'' ≡ 1 (mod 3) ⇔ ''R''<sub>''n''</sub> ≡ 1 (mod 3) ⇔ ''R''<sub>''n''</sub> ≡ ''R''<sub>1</sub> ≡ 1 (mod ''R''<sub>3</sub>),<br />''n'' ≡ 2 (mod 3) ⇔ ''R''<sub>''n''</sub> ≡ 2 (mod 3) ⇔ ''R''<sub>''n''</sub> ≡ ''R''<sub>2</sub> ≡ 11 (mod ''R''<sub>3</sub>).<br />Therefore, 3 | ''n'' ⇔ 3 | ''R''<sub>''n''</sub> ⇔ ''R''<sub>3</sub> | ''R''<sub>''n''</sub>.
* The remainder of ''R''<sub>''n''</sub> modulo 9 is equal to the remainder of ''n'' modulo 9. Using 10<sup>''a''</sup> ≡ 1 (mod 9) for any ''a'' &ge; 0,<br />''n'' ≡ ''r'' (mod 9) ⇔ ''R''<sub>''n''</sub> ≡ ''r'' (mod 9) ⇔ ''R''<sub>''n''</sub> ≡ ''R''<sub>''r''</sub> (mod ''R''<sub>9</sub>),<br />for 0 &le; ''r'' &lt; 9.<br />Therefore, 9 | ''n'' ⇔ 9 | ''R''<sub>''n''</sub> ⇔ ''R''<sub>9</sub> | ''R''<sub>''n''</sub>.

=== Algebra factorization of generalized 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 latter 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.

=== The generalized repunit conjecture ===

A conjecture related to the generalized repunit primes:<ref>[http://primes.utm.edu/mersenne/heuristic.html Deriving the Wagstaff Mersenne Conjecture]</ref><ref>[https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0906&L=NMBRTHRY&P=R295&1=NMBRTHRY&9=A&J=on&d=No+Match%3BMatch%3BMatches&z=4 Generalized Repunit Conjecture]</ref> (the conjecture predicts where is the next [[generalized Mersenne prime]], if the conjecture is true, then there are infinitely many repunit primes for all bases <math>b</math>)

For any integer <math>b</math>, which satisfies the conditions:

# <math>|b|>1</math>.
# <math>b</math> is not a [[perfect power]]. (since when <math>b</math> is a perfect <math>r</math>th power, it can be shown that there is at most one <math>n</math> value such that <math>\frac{b^n-1}{b-1}</math> is prime, and this <math>n</math> value is <math>r</math> itself or a [[nth root|root]] of <math>r</math>)
# <math>b</math> is not in the form <math>-4k^4</math>. (if so, then the number has [[aurifeuillean factorization]])

has generalized repunit primes of the form

:<math>R_p(b)=\frac{b^p-1}{b-1}</math>

for prime <math>p</math>, the prime numbers will be distributed near the best fit line

:<math>
Y=G \cdot \log_{|b|}\left( \log_{|b|}\left( R_{(b)}(n) \right) \right)+C,
</math>

where limit <math>n\rightarrow\infty</math>, <math>G=\frac{1}{e^\gamma}=0.561459483566...</math>

and there are about

:<math>
\left( \log_e(N)+m \cdot \log_e(2) \cdot \log_e \big( \log_e(N) \big)
+\frac{1}{\sqrt N}-\delta \right) \cdot \frac{e^\gamma}{\log_e(|b|)}
</math>

base-''b'' repunit primes less than ''N''.

*<math>e</math> is the [[e (mathematical constant)|base of natural logarithm]].
*<math>\gamma</math> is [[Euler–Mascheroni constant]].
*<math>\log_{|b|}</math> is the [[logarithm]] in [[base of a logarithm|base]] <math>|b|</math>
*<math>R_{(b)}(n)</math> is the <math>n</math>th generalized repunit prime in base''b'' (with prime ''p'')
*<math>C</math> is a data fit constant which varies with <math>b</math>.
*<math>\delta=1</math> if <math>b>0</math>, <math>\delta=1.6</math> if <math>b<0</math>.
*<math>m</math> is the largest natural number such that <math>-b</math> is a <math>2^{m-1}</math>th power.

We also have the following 3 properties:

# The number of prime numbers of the form <math>\frac{b^n-1}{b-1}</math> (with prime <math>p</math>) less than or equal to <math>n</math> is about <math>e^\gamma \cdot \log_{|b|}\big(\log_{|b|}(n)\big)</math>.
# The expected number of prime numbers of the form <math>\frac{b^n-1}{b-1}</math> with prime <math>p</math> between <math>n</math> and <math>|b| \cdot n</math> is about <math>e^\gamma</math>. 
# The probability that number of the form <math>\frac{b^n-1}{b-1}</math> is prime (for prime <math>p</math>) is about <math>\frac{e^\gamma}{p \cdot \log_e(|b|)}</math>.