Repunit

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by A.H. Beiler. A repunit prime is a repunit that is also a prime number.

Definition

The repunits are defined mathematically as:$$R_n={10^n-1over9}qquadmbox{for }nge1.Thus, the number "R"_"n" consists of "n" copies of the digit 1. The sequence of repunits starts 1, 11, 111, 1111,... (sequence in OEIS).

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 recurring decimals [Dickson, Leonard Eugene and Cresse, G.H.; "History of the Theory of Numbers"; pp. 164-167 ISBN 0821819348] .

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 [Ibid] and it is curious that, though Edouard 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 [Francis, Richard L.; "Mathematical Haystacks: Another Look at Repunit Numbers" in The College Mathematics Journal, Vol. 19, No. 3. (May, 1988), pp. 240-246.] 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.

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" is divisible by "R"_"a":

:$$R_n=frac{1}{9}prod_{d|n}Phi_d(10)

where $$Phi_d is the $$d^mathrm{th} cyclotomic polynomial and "d" ranges over the divisors of "n". For "p" prime, $$Phi_p(x)=sum_{i=0}^{p-1}x^i, which has the expected form of a repunit when "x" is substituted for with 10.

For example, 9 is divisible by 3, and indeed "R"₉ is divisible by "R"₃—in fact, 111111111 = 111 · 1001001. The corresponding cyclotomic polynomals $$Phi_3(x) and $$Phi_9(x) are $$x^2+x+1 and $$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, "R"₃ = 111 = 3 · 37 is not prime. Except for this case of "R"₃, "p" can only divide "R"_"n" for prime "n" if "p = 2kn + 1" for some "k".

"R"_"n" is prime for "n" = 2, 19, 23, 317, 1031,... (sequence in OEIS). "R"₄₉₀₈₁ and "R"₈₆₄₅₃ are probably prime. On April 3 2007 Harvey Dubner (who also found "R"₄₉₀₈₁) announced that "R"₁₀₉₂₉₇ is a probable prime. [Harvey Dubner, " [http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0704&L=nmbrthry&T=0&P=178 New Repunit R(109297)] "] He later announced there are no others from "R"₈₆₄₅₃ to "R"₂₀₀₀₀₀. [Harvey Dubner, " [http://tech.groups.yahoo.com/group/primeform/message/8546 Repunit search limit] "] On July 15 2007 Maksym Voznyy announced "R"₂₇₀₃₄₃ to be probably prime [Maksym Voznyy, " [http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0707&L=nmbrthry&T=0&P=1086 New PRP Repunit R(270343)] "] , along with his intent to search to 400000.

It has been conjectured that there are infinitely many repunit primes [http://primes.utm.edu/glossary/page.php?sort=Repunit] 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 primes, i.e., primes that remain prime after any permutation of their digits.

Generalizations

Professional mathematicians used to consider repunits an arbitrary concept, since they depend on the use of decimal numerals. But the arbitrariness can be removed by generalizing the idea to base-"b" repunits::$$R_n^{(b)}={b^n-1over b-1}qquadmbox{for }nge1.

In fact, the base-2 repunits are the well-respected Mersenne numbers "M"_"n" = 2^"n" − 1. 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.

Example 1) the first few base-3 repunit primes are 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (sequence in OEIS), corresponding to $$n of 3, 7, 13, 71, 103 (sequence in OEIS).

Example 2) the only base-4 repunit prime is 5 ($$11_4), because $$4^n-1=left(2^n+1 ight)left(2^n-1 ight), and 3 divides one of these, leaving the other as a factor of the repunit.

It is easy to [http://www.caliban.org.uk/pmwiki/pmwiki.php?n=Blogs.RichardRothwell.RepUnits prove] that given "n", such that "n" is not exactly divisible by 2 or "p", there exists a repunit in base 2"p" that is a multiple of "n".

ee also

* Repdigit
* Recurring decimal
* All one polynomial - Another generalization
* Goormaghtigh conjecture

References

External links

Web sites
*mathworld|urlname=Repunit|title=Repunit
* [http://www.cerias.purdue.edu/homes/ssw/cun/third/pmain901 The main tables] of the [http://www.cerias.purdue.edu/homes/ssw/cun/ Cunningham project] .
* [http://primes.utm.edu/glossary/page.php?sort=Repunit Repunit] at [http://primes.utm.edu/ The Prime Pages] by Chris Caldwell.
* [http://www.worldofnumbers.com/repunits.htm Repunits and their prime factors] at [http://www.worldofnumbers.com World!Of Numbers] .Books
*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.