Inrecreational mathematics, arepunit is anumber like 11, 111, or 1111 that contains only the digit1 — a more specific type ofrepdigit. The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his bookRecreations in the Theory of Numbers.^{[note 1]}

Arepunit prime is a repunit that is also aprime number. Primes that are repunits inbase-2 areMersenne primes. As of May 2025, thelargest known prime number2^136,279,841 − 1, the largestprobable primeR_8177207 and the largestelliptic curve primality-proven primeR₁₀₉₂₉₇ are all repunits in various bases.

The base-b repunits are defined as (thisb can be either positive or negative)

${\displaystyle R_n^{(b)}\equiv 1+b+b^2+\cdots +b^{n-1}=\frac{b^n-1}{b-1}\text{for }|b|\ge 2,n\ge 1.}$

Thus, the numberR_n^(b) consists ofn copies of the digit 1 in base-b representation. The first two repunits base-b forn = 1 andn = 2 are

${\displaystyle R_1^{(b)}=\frac{b-1}{b-1}=1\text{and}{R}_2^{(b)}=\frac{b^2-1}{b-1}=b+1\text{for}|b|\ge 2.}$

In particular, thedecimal (base-10) repunits that are often referred to as simplyrepunits are defined as

${\displaystyle R_n\equiv R_n^{(10)}=\frac{{10}^n-1}{10-1}=\frac{{10}^n-1}{9}\text{for }n\ge 1.}$

Thus, the numberR_n =R_n⁽¹⁰⁾ consists ofn copies of the digit 1 in base 10 representation. The sequence of repunits base-10 starts with

1,11,111, 1111, 11111, 111111, ... (sequenceA002275 in theOEIS).

Similarly, the repunits base-2 are defined as

${\displaystyle R_n^{(2)}=\frac{2^n-1}{2-1}=2^n-1\text{for }n\ge 1.}$

Thus, the numberR_n⁽²⁾ consists ofn copies of the digit 1 in base-2 representation. In fact, the base-2 repunits are the well-knownMersenne numbersM_n = 2ⁿ − 1, they start with

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, ... (sequenceA000225 in theOEIS).

• Any repunit in any base having a composite number of digits is necessarily composite. For example,

  R₃₅^(b) = 11111111111111111111111111111111111 = 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.

Only repunits (in any base) having a prime number of digits can be prime. This is anecessary but not sufficient condition. For example,

R₁₁⁽²⁾ = 2¹¹ − 1 = 2047 = 23 × 89.

• Ifp is an odd prime, then every primeq that dividesR_p^(b) must be either 1 plus a multiple of 2p, or a factor ofb − 1. For example, a prime factor ofR₂₉ is 62003 = 1 + 2·29·1069. The reason is that the primep is the smallest exponent greater than 1 such thatq dividesb^p − 1, becausep is prime. Therefore, unlessq dividesb − 1,p divides theCarmichael function ofq, which is even and equal toq − 1.
• Any positive multiple of the repunitR_n^(b) contains at leastn nonzero digits in base-b.
• Any numberx is a two-digit repunit in base x − 1.
• 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). TheGoormaghtigh conjecture says there are only these two cases.
• Using thepigeon-hole principle it can be easily shown that forrelatively prime natural numbersn andb, there exists a repunit in base-b that is a multiple ofn. To see this consider repunitsR₁^(b),...,R_n^(b). Because there aren repunits but onlyn−1 non-zero residues modulon there exist two repunitsR_i^(b) andR_j^(b) with 1 ≤i <j ≤n such thatR_i^(b) andR_j^(b) have the same residue modulon. It follows thatR_j^(b) −R_i^(b) has residue 0 modulon, i.e. is divisible byn. SinceR_j^(b) −R_i^(b) consists ofj −i ones followed byi zeroes,R_j^(b) −R_i^(b) =R_j−i^(b) ×bⁱ. Nown divides the left-hand side of this equation, so it also divides the right-hand side, but sincen andb are relatively prime,n must divideR_j−i^(b).
• TheFeit–Thompson conjecture is thatR_q^(p) never dividesR_p^(q) for two distinct primesp andq.
• Using theEuclidean Algorithm for repunits definition:R₁^(b) = 1;R_n^(b) =R_n−1^(b) ×b + 1, any consecutive repunitsR_n−1^(b) andR_n^(b) are relatively prime in any base-b for anyn.
• Ifm andn have a common divisord,R_m^(b) andR_n^(b) have the common divisorR_d^(b) in any base-b for anym andn. That is, the repunits of a fixed base form astrong divisibility sequence. As a consequence, Ifm andn are relatively prime,R_m^(b) andR_n^(b) are relatively prime. The Euclidean Algorithm is based ongcd(m,n) =gcd(m −n,n) form >n. Similarly, usingR_m^(b) −R_n^(b) ×b^m−n =R_m−n^(b), it can be easily shown thatgcd(R_m^(b),R_n^(b)) =gcd(R_m−n^(b),R_n^(b)) form >n. Therefore, ifgcd(m,n) =d, thengcd(R_m^(b),R_n^(b)) =R_d^(b).

(Prime factors (or prime powers) parenthesized and colored(red) are "new factors", i. e. the prime factor (or power) dividesR_n but does not divideR_k for allk <n) (sequenceA102380 in theOEIS)^[2]

Smallest prime factor ofR_n forn > 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, ... (sequenceA067063 in theOEIS)

The definition of repunits was motivated by recreational mathematicians looking forprime factors of such numbers.

It is easy to show that ifn is divisible bya, thenR_n^(b) is divisible byR_a^(b):

${\displaystyle R_n^{(b)}=\frac{1}{b-1}\prod _{d|n}{\mathrm{\Phi }}_d(b),}$

where${\displaystyle {\mathrm{\Phi }}_d(x)}$ is the${\displaystyle d^{\mathrm{t}\mathrm{h}}}$cyclotomic polynomial andd ranges over the divisors ofn. Forp prime,

${\displaystyle {\mathrm{\Phi }}_p(x)=\sum _{i=0}^{p-1}{x}^i,}$

which has the expected form of a repunit whenx is substituted withb.

For example, 9 is divisible by 3, and thusR₉ is divisible byR₃—in fact, 111111111 = 111 · 1001001. The corresponding cyclotomic polynomials${\displaystyle {\mathrm{\Phi }}_3(x)}$ and${\displaystyle {\mathrm{\Phi }}_9(x)}$ are${\displaystyle x^2+x+1}$ and${\displaystyle x^6+x^3+1}$, respectively. Thus, forR_n to be prime,n must necessarily be prime, but it is not sufficient forn to be prime. For example,R₃ = 111 = 3 · 37 is not prime. Except for this case ofR₃,p can only divideR_n for primen ifp = 2kn + 1 for somek.

R_n is prime forn = 2, 19, 23, 317, 1031, 49081, 86453, 109297 ... (sequenceA004023 inOEIS). On July 15, 2007, Maksym Voznyy announcedR₂₇₀₃₄₃ to be probably prime.^[3] Serge Batalov and Ryan Propper foundR_5794777 andR_8177207 to be probable primes on April 20 and May 8, 2021, respectively.^[4] As of their discovery, each was the largest known probable prime. On March 22, 2022, probable primeR₄₉₀₈₁ was eventually proven to be a prime.^[5] On May 15, 2023, probable primeR₈₆₄₅₃ was eventually proven to be a prime.^[6] On May 26, 2025, probable primeR₁₀₉₂₉₇ was eventually proven to be a prime.^[7]

It has been conjectured that there are infinitely many repunit primes^[8] and they seem to occur roughly as often as theprime number theorem would predict: the exponent of theNth repunit prime is generally around a fixed multiple of the exponent of the (N−1)th.

The prime repunits are a trivial subset of thepermutable primes, i.e., primes that remain prime after anypermutation of their digits.

Particular properties are

• The remainder ofR_n modulo 3 is equal to the remainder ofn modulo 3. Using 10^a ≡ 1 (mod 3) for anya ≥ 0,
  n ≡ 0 (mod 3) ⇔R_n ≡ 0 (mod 3) ⇔R_n ≡ 0 (modR₃),
  n ≡ 1 (mod 3) ⇔R_n ≡ 1 (mod 3) ⇔R_n ≡R₁ ≡ 1 (modR₃),
  n ≡ 2 (mod 3) ⇔R_n ≡ 2 (mod 3) ⇔R_n ≡R₂ ≡ 11 (modR₃).
  Therefore, 3 |n ⇔ 3 |R_n ⇔R₃ |R_n.
• The remainder ofR_n modulo 9 is equal to the remainder ofn modulo 9. Using 10^a ≡ 1 (mod 9) for anya ≥ 0,
  n ≡r (mod 9) ⇔R_n ≡r (mod 9) ⇔R_n ≡R_r (modR₉),
  for 0 ≤r < 9.
  Therefore, 9 |n ⇔ 9 |R_n ⇔R₉ |R_n.

Ifb is aperfect power (can be written asmⁿ, withm,n integers,n > 1) differs from 1, then there is at most one repunit in base-b. Ifn is aprime power (can be written asp^r, withp prime,r integer,p,r >0), then all repunit in base-b are not prime aside fromR_p andR₂.R_p 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., andR₂ can be prime (whenp differs from 2) only ifb is negative, a power of −2, for example,b = −8, −32, −128, −8192, etc., in fact, theR₂ can also be composite, for example,b = −512, −2048, −32768, etc. Ifn is not a prime power, then no base-b repunit prime exists, for example,b = 64, 729 (withn = 6),b = 1024 (withn = 10), andb = −1 or 0 (withn any natural number). Another special situation isb = −4k⁴, withk positive integer, which has theaurifeuillean factorization, for example,b = −4 (withk = 1, thenR₂ andR₃ are primes), andb = −64, −324, −1024, −2500, −5184, ... (withk = 2, 3, 4, 5, 6, ...), then no base-b repunit prime exists. It is also conjectured that whenb is neither a perfect power nor −4k⁴ withk positive integer, then there are infinity many base-b repunit primes.

A conjecture related to the generalized repunit primes:^[9][10] (the conjecture predicts where is the nextgeneralized Mersenne prime, if the conjecture is true, then there are infinitely many repunit primes for all bases${\displaystyle b}$)

For any integer${\displaystyle b}$, which satisfies the conditions:

1. ${\displaystyle |b|>1}$.
2. ${\displaystyle b}$ is not aperfect power. (since when${\displaystyle b}$ is a perfect${\displaystyle r}$th power, it can be shown that there is at most one${\displaystyle n}$ value such that${\displaystyle \frac{b^n-1}{b-1}}$ is prime, and this${\displaystyle n}$ value is${\displaystyle r}$ itself or aroot of${\displaystyle r}$)
3. ${\displaystyle b}$ is not in the form${\displaystyle -4k^4}$. (if so, then the number hasaurifeuillean factorization)

has generalized repunit primes of the form

${\displaystyle R_p(b)=\frac{b^p-1}{b-1}}$

for prime${\displaystyle p}$, the prime numbers will be distributed near the best fit line

${\displaystyle Y=G\cdot {\log }_{|b|}\left({\log }_{|b|}\left(R_{(b)}(n)\right)\right)+C,}$

where limit${\displaystyle n\to \mathrm{\infty }}$,${\displaystyle G=\frac{1}{e^{\gamma }}=\mathrm{0.561459483566...}}$

and there are about

${\displaystyle \left({\log }_e(N)+m\cdot {\log }_e(2)\cdot {\log }_e({\log }_e(N))+\frac{1}{\sqrt{N}}-\delta \right)\cdot \frac{e^{\gamma }}{{\log }_e(|b|)}}$

base-b repunit primes less thanN.

• ${\displaystyle e}$ is thebase of natural logarithm.
• ${\displaystyle \gamma }$ isEuler–Mascheroni constant.
• ${\displaystyle {\log }_{|b|}}$ is thelogarithm inbase${\displaystyle |b|}$
• ${\displaystyle R_{(b)}(n)}$ is the${\displaystyle n}$th generalized repunit prime in baseb (with primep)
• ${\displaystyle C}$ is a data fit constant which varies with${\displaystyle b}$.
• ${\displaystyle \delta =1}$ if${\displaystyle b>0}$,${\displaystyle \delta =1.6}$ if.
• ${\displaystyle m}$ is the largest natural number such that${\displaystyle -b}$ is a${\displaystyle 2^{m-1}}$th power.

We also have the following 3 properties:

1. The number of prime numbers of the form${\displaystyle \frac{b^n-1}{b-1}}$ (with prime${\displaystyle p}$) less than or equal to${\displaystyle n}$ is about${\displaystyle e^{\gamma }\cdot {\log }_{|b|}({\log }_{|b|}(n))}$.
2. The expected number of prime numbers of the form${\displaystyle \frac{b^n-1}{b-1}}$ with prime${\displaystyle p}$ between${\displaystyle n}$ and${\displaystyle |b|\cdot n}$ is about${\displaystyle e^{\gamma }}$.
3. The probability that number of the form${\displaystyle \frac{b^n-1}{b-1}}$ is prime (for prime${\displaystyle p}$) is about${\displaystyle \frac{e^{\gamma }}{p\cdot {\log }_e(|b|)}}$.

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 ofrepeating decimals.^[11]

It was found very early on that for any primep greater than 5, theperiod of the decimal expansion of 1/p is equal to the length of the smallest repunit number that is divisible byp. Tables of the period of reciprocal of primes up to 60,000 had been published by 1860 and permitted thefactorization by such mathematicians as Reuschle of all repunits up toR₁₆ and many larger ones. By 1880, evenR₁₇ toR₃₆ had been factored^[11] and it is curious that, thoughÉdouard Lucas showed no prime below three million had periodnineteen, there was no attempt to test any repunit for primality until early in the twentieth century. The American mathematician Oscar Hoppe provedR₁₉ to be prime in 1916,^[12] and Lehmer and Kraitchik independently foundR₂₃ 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.R₃₁₇ was found to be aprobable prime circa 1966 and was proved prime eleven years later, whenR₁₀₃₁ 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.

TheCunningham project endeavours to document the integer factorizations of (among other numbers) the repunits to base 2, 3, 5, 6, 7, 10, 11, and 12.

D. R. Kaprekar has defined Demlo numbers as concatenation of a left, middle and right part, where the left and right part must be of the same length (up to a possible leading zero to the left) and must add up to a repdigit number, and the middle part may contain any additional number of this repeated digit.^[13] They are named afterDemlo railway station (now calledDombivili) 30 miles from Bombay on the thenG.I.P. Railway, where Kaprekar started investigating them.He callsWonderful Demlo numbers those of the form 1, 121, 12321, 1234321, ..., 12345678987654321. The fact that these are the squares of the repunits has led some authors to call Demlo numbers the infinite sequence of these,^[14] 1, 121, 12321, ..., 12345678987654321, 1234567900987654321, 123456790120987654321, ..., (sequenceA002477 in theOEIS), although one can check these are not Demlo numbers forp = 10, 19, 28, ...

• All one polynomial — Another generalization
• Goormaghtigh conjecture
• Repeating decimal
• Repdigit
• Wagstaff prime — can be thought of as repunit primes withnegative base${\displaystyle b=-2}$

• Beiler, Albert H. (2013) [1964],Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover Recreational Math (2nd Revised ed.), New York: Dover Publications,ISBN 978-0-486-21096-4
• Dickson, Leonard Eugene;Cresse, G.H. (1999),History of the Theory of Numbers, Volume I: Divisibility and primality (2nd Reprinted ed.), Providence, RI: AMS Chelsea Publishing,ISBN 978-0-8218-1934-0
• Francis, Richard L. (1988), "Mathematical Haystacks: Another Look at Repunit Numbers",The College Mathematics Journal,19 (3):240–246,doi:10.1080/07468342.1988.11973120
• Gunjikar, K. R.;Kaprekar, D. R. (1939),"Theory of Demlo numbers"(PDF),Journal of the University of Bombay,VIII (3):3–9
• Kaprekar, D. R. (1938a),"On Wonderful Demlo numbers",The Mathematics Student,6: 68
• Kaprekar, D. R. (1938b), "Demlo numbers",J. Phys. Sci. Univ. Bombay,VII (3)
• Kaprekar, D. R. (1948),Demlo numbers, Devlali, India: Khareswada
• Ribenboim, Paulo (1996-02-02),The New Book of Prime Number Records, Computers and Medicine (3rd ed.), New York: Springer,ISBN 978-0-387-94457-9
• Yates, Samuel (1982),Repunits and repetends, FL: Delray Beach,ISBN 978-0-9608652-0-8

• Weisstein, Eric W."Repunit".MathWorld.
• The main tables of theCunningham project.
• Repunit atThe Prime Pages by Chris Caldwell.
• Repunits and their prime factors atWorld!Of Numbers.
• Prime generalized repunits of at least 1000 decimal digits by Andy Steward
• Repunit Primes Project Giovanni Di Maria's repunit primes page.
• Smallest odd prime p such that (b^p-1)/(b-1) and (b^p+1)/(b+1) is prime for bases 2<=b<=1024
• Factorization of repunit numbers
• Generalized repunit primes in base -50 to 50

• Integers
• Base-dependent integer sequences

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