Editing Repunit (section)

=== 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>.