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 repeated unit and was coined in 1966 by Albert H. Beiler.
- 1 Definition
- 2 Properties
- 3 Factorization of decimal repunits
- 4 Repunit primes
- 5 History
- 6 Demlo numbers
- 7 See also
- 8 Notes
- 9 External links
The base-b repunits are defined as
Thus, the number Rn(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
In particular, the decimal (base-10) repunits that are often referred to as simply repunits are defined as
Thus, the number Rn = Rn(10) consists of n copies of the digit 1 in base 10 representation. The sequence of repunits base 10 starts with
Similarly, the repunits base 2 are defined as
Thus, the number Rn(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 Mn = 2n − 1.
- 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.
- Any positive multiple of the repunit Rn(b) contains at least n nonzero digits in base b.
- 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 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 R1(b),...,Rn(b). Assume none of the Rk(b) is divisible by n. Because there are n repunits but only n-1 non-zero residues modulo n there exist two repunits Ri(b) and Rj(b) with 1≤i<j≤n such that Ri(b) and Rj(b) have the same residue modulo n. It follows that Rj(b) - Ri(b) has residue 0 modulo n, i.e. is divisible by n. Rj(b) - Ri(b) consists of j - i ones followed by i zeroes. Thus, Rj(b) - Ri(b) = Rj-i(b) x 10i = Rj-i(b) x bi . 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 Rj-i(b) contradicting the original assumption.
- The Feit–Thompson conjecture is that Rq(p) never divides Rp(q) for two distinct primes p and q.
Factorization of decimal repunits
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 Rn(b) is divisible by Ra(b):
where is the cyclotomic polynomial and d ranges over the divisors of n. For p prime, , 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 and are and respectively. Thus, for Rn 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 Rn for prime n if p = 2kn + 1 for some k.
Decimal repunit primes
Rn is prime for n = 2, 19, 23, 317, 1031,... (sequence A004023 in OEIS). R49081 and R86453 are probably prime. On April 3, 2007 Harvey Dubner (who also found R49081) announced that R109297 is a probable prime. He later announced there are no others from R86453 to R200000. On July 15, 2007 Maksym Voznyy announced R270343 to be probably prime, along with his intent to search to 400000. As of November 2012, all further candidates up to R2500000 have been tested, but no new probable primes have been found so far.
It has been conjectured that there are infinitely many repunit primes and they seem to occur roughly as often as the prime number theorem would predict: the exponent of the Nth repunit prime is generally around a fixed multiple of the exponent of the (N-1)th.
Base-2 repunit primes
Base-2 repunit primes are called Mersenne primes.
Base-3 repunit primes
The first few base-3 repunit primes are
- 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013, ... (sequence A076481 in OEIS),
corresponding to of
Base-4 repunit primes
The only base-4 repunit prime is 5 (). , and 3 always divides when n is odd and when n is even. For n greater than 2, both and are greater than 3, so removing the factor of 3 still leaves two factors greater than 1, so the number cannot be prime.
Base 5 repunit primes
The first few base-5 (quinary) repunit primes are
corresponding to of
Base 6 repunit primes
The first few base-6 repunit primes are
- 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371, ..., (sequence A165210 in OEIS)
corresponding to of
Base 7 repunit primes
The first few base 7 repunit primes are
- 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457,
corresponding to of
Base 8 and 9 repunit primes
Base 12 (duodecimal) repunit primes
The first few base 12 repunit primes are
- 13, 157, 22621, 29043636306420266077, 435700623537534460534556100566709740005056966111842089407838902783209959981593077811330507328327968191581, 388475052482842970801320278964160171426121951256610654799120070705613530182445862582590623785872890159937874339918941
corresponding to of
The only known vigesimal (base 20) repunit primes or probable primes are for of
The first three of these in decimal are
- 421, 10778947368421 and 689852631578947368421
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 decimals.
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 had been factored 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 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.
The Demlo numbers 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.
- Harvey Dubner, New Repunit R(109297)
- Harvey Dubner, Repunit search limit
- Maksym Voznyy, New PRP Repunit R(270343)
- Chris Caldwell, "The Prime Glossary: repunit" at The Prime Pages.
- Dickson, Leonard Eugene and Cresse, G.H.; History of the Theory of Numbers; pp. 164-167 ISBN 0-8218-1934-8
- Dickson and Cresse, pp. 164-167
- Francis, Richard L.; "Mathematical Haystacks: Another Look at Repunit Numbers" in The College Mathematics Journal, Vol. 19, No. 3. (May, 1988), pp. 240-246.
- Weisstein, Eric W., "Demlo Number", MathWorld.
- 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.
- Factorizations of 11...11 (Repunit) by Makoto Kamada
- S. Yates, Repunits and repetends. ISBN 0-9608652-0-9.
- A. Beiler, Recreations in the theory of numbers. ISBN 0-486-21096-0. Chapter 11, of course.
- Paulo Ribenboim, The New Book Of Prime Number Records. ISBN 0-387-94457-5.