In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.^{[note 1]}

A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes. As of May 2023, the largest known prime number 2^82,589,933 − 1, the largest probable prime R_8177207 and the largest elliptic curve primality-proven prime R₈₆₄₅₃ are all repunits in various bases.

Definition

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

${\displaystyle R_{n}^{(b)}\equiv 1+b+b^{2}+\cdots +b^{n-1}={b^{n}-1 \over {b-1}}\qquad {\mbox{for }}|b|\geq 2,n\geq 1.}$

Thus, the number R_n^(b) 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

${\displaystyle R_{1}^{(b)}={b-1 \over {b-1}}=1\qquad {\text{and}}\qquad R_{2}^{(b)}={b^{2}-1 \over {b-1}}=b+1\qquad {\text{for}}\ |b|\geq 2.}$

In particular, the decimal (base-10) repunits that are often referred to as simply repunits are defined as

${\displaystyle R_{n}\equiv R_{n}^{(10)}={10^{n}-1 \over {10-1}}={10^{n}-1 \over 9}\qquad {\mbox{for }}n\geq 1.}$

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

1, 11, 111, 1111, 11111, 111111, ... (sequence A002275 in the OEIS).

Similarly, the repunits base-2 are defined as

${\displaystyle R_{n}^{(2)}={2^{n}-1 \over {2-1}}={2^{n}-1}\qquad {\mbox{for }}n\geq 1.}$

Thus, the number R_n⁽²⁾ consists of n copies of the digit 1 in base-2 representation. In fact, the base-2 repunits are the well-known Mersenne numbers M_n = 2ⁿ − 1, they start with

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, ... (sequence A000225 in the OEIS).

Properties

• 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,

  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.

• If p is an odd prime, then every prime q that divides R_p^(b) must be either 1 plus a multiple of 2p, or a factor of b − 1. For example, a prime factor of R₂₉ is 62003 = 1 + 2·29·1069. The reason is that the prime p is the smallest exponent greater than 1 such that q divides b^p − 1, because p is prime. Therefore, unless q divides b − 1, p divides the Carmichael function of q, which is even and equal to q − 1.
• Any positive multiple of the repunit R_n^(b) contains at least n nonzero digits in base-b.
• Any number x 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). The Goormaghtigh conjecture says there are only these two cases.
• Using the pigeon-hole principle it can be easily shown that for relatively prime natural numbers n and b, there exists a repunit in base-b that is a multiple of n. To see this consider repunits R₁^(b),...,R_n^(b). Because there are n repunits but only n−1 non-zero residues modulo n there exist two repunits R_i^(b) and R_j^(b) with 1 ≤ i < j ≤ n such that R_i^(b) and R_j^(b) have the same residue modulo n. It follows that R_j^(b) − R_i^(b) has residue 0 modulo n, i.e. is divisible by n. Since R_j^(b) − R_i^(b) consists of j − i ones followed by i zeroes, R_j^(b) − R_i^(b) = R_j−i^(b) × bⁱ. Now n divides the left-hand side of this equation, so it also divides the right-hand side, but since n and b are relatively prime, n must divide R_j−i^(b).
• The Feit–Thompson conjecture is that R_q^(p) never divides R_p^(q) for two distinct primes p and q.
• Using the Euclidean Algorithm for repunits definition: R₁^(b) = 1; R_n^(b) = R_n−1^(b) × b + 1, any consecutive repunits R_n−1^(b) and R_n^(b) are relatively prime in any base-b for any n.
• If m and n have a common divisor d, R_m^(b) and R_n^(b) have the common divisor R_d^(b) in any base-b for any m and n. That is, the repunits of a fixed base form a strong divisibility sequence. As a consequence, If m and n are relatively prime, R_m^(b) and R_n^(b) are relatively prime. The Euclidean Algorithm is based on gcd(m, n) = gcd(m − n, n) for m > n. Similarly, using R_m^(b) − R_n^(b) × b^m−n = R_m−n^(b), it can be easily shown that gcd(R_m^(b), R_n^(b)) = gcd(R_m−n^(b), R_n^(b)) for m > n. Therefore, if gcd(m, n) = d, then gcd(R_m^(b), R_n^(b)) = R_d^(b).

Factorization of decimal repunits

(Prime factors colored red means "new factors", i. e. the prime factor divides R_n but does not divide R_k for all k < n) (sequence A102380 in the OEIS)^[2]

Smallest prime factor of R_n 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, ... (sequence A067063 in the OEIS)

Repunit primes

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

It is easy to show that if n is divisible by a, then R_n^(b) is divisible by R_a^(b):

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

where ${\displaystyle \Phi _{d}(x)}$ is the ${\displaystyle d^{\mathrm {th} }}$ cyclotomic polynomial and d ranges over the divisors of n. For p prime,

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

which has the expected form of a repunit when x is substituted with b.

For example, 9 is divisible by 3, and thus R₉ is divisible by R₃—in fact, 111111111 = 111 · 1001001. The corresponding cyclotomic polynomials ${\displaystyle \Phi _{3}(x)}$ and ${\displaystyle \Phi _{9}(x)}$ are ${\displaystyle x^{2}+x+1}$ and ${\displaystyle x^{6}+x^{3}+1}$, respectively. Thus, for R_n to be prime, n must necessarily be prime, but it is not sufficient for n to be prime. For example, R₃ = 111 = 3 · 37 is not prime. Except for this case of R₃, p can only divide R_n for prime n if p = 2kn + 1 for some k.

History

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 repeating decimals.^[11]

It was found very early on that for any prime p greater than 5, the 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 factorization by such mathematicians as Reuschle of all repunits up to R₁₆ and many larger ones. By 1880, even R₁₇ to R₃₆ had been factored^[11] 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 R₁₉ to be prime in 1916^[12] and Lehmer and Kraitchik independently found R₂₃ 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 a probable prime circa 1966 and was proved prime eleven years later, when R₁₀₃₁ 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.

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.

Demlo numbers

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 after Demlo railway station (now called Dombivili) 30 miles from Bombay on the then G.I.P. Railway, where Kaprekar started investigating them. He calls Wonderful 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, ..., (sequence A002477 in the OEIS), although one can check these are not Demlo numbers for p = 10, 19, 28, ...

See also

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

Footnotes

References

• 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

External links

• Weisstein, Eric W. "Repunit". MathWorld.
• The main tables of the Cunningham project.
• Repunit at The Prime Pages by Chris Caldwell.
• Repunits and their prime factors at World!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