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Encyclopedia > 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. Recreational mathematics includes many mathematical games, and can be extended to cover such areas as logic and other puzzles of deductive reasoning. ... A number is an abstract entity that represents a count or measurement. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...

Contents

Definition

The repunits are defined mathematically as

R_n={10^n-1over9}qquadmbox{for }nge1.

Thus, the number Rn consists of n copies of the digit 1. The sequence of repunits starts 111111, 1111,... (sequence A002275 in OEIS). Look up one in Wiktionary, the free dictionary. ... 11 (eleven) is the natural number following 10 and preceding 12. ... 111 is the natural number following 110 and preceding 112. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...


Repunit primes

Historically, the definition of repunits was motivated by recreational mathematicians looking for prime factors of such numbers. In number theory, the integer factorization problem is the problem of finding a non-trivial divisor of a composite number; for example, given a number like 91, the challenge is to find a number such as 7 which divides it. ...


It is easy to show that if n is divisible by a, then Rn is divisible by Ra:

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

where Φd is the dth cyclotomic polynomial. For p prime, Phi_p(x)=sum_{i=0}^{p-1}x^i, which has the expected form of a repunit when x is subsituted for with 10. In mathematics, the nth roots of unity or de Moivre numbers are all the complex numbers which yield 1 when raised to a given power n. ...


For example, 9 is divisible by 3, and indeed R9 is divisible by R3—in fact, 111111111 = 111 · 1001001. The corresponding cyclotomic polynomals Φ3(x) and Φ9(x) are x2 + x + 1 and x6 + x3 + 1 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 if p = 2kn + 1 for some k.


Repunit primes turn out to be rare. 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.[1] It has been conjectured that there are infinitely many repunit primes. The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ... In number theory, a probable prime (PRP) is an integer that satisfies a condition also satisfied by all prime numbers. ... April 3 is the 93rd day of the year (94th in leap years) in the Gregorian calendar, with 272 days remaining. ... 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the Anno Domini (common) era. ...


The prime repunits are a subset of the permutable primes, i.e., primes that remain prime after any permutation of their digits. A permutable prime (sometimes called a primutation) is a prime number, which, in a given base, can have its digits switched to any possible permutation and still spell a prime number. ... Template:Hellodablink Permutation is the rearrangement of objects or symbols into distinguishable sequences. ...


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: The decimal (base ten or occasionally denary) numeral system has ten as its base. ...

R_n^{(b)}={b^n-1over b-1}qquadmbox{for }nge1.

In fact, the base-2 repunits are the well-respected Mersenne numbers Mn = 2n − 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. In mathematics, a Mersenne number is a number that is one less than a power of two. ... The Cunningham project aims to find factors of large numbers of the form bn ± 1 for b = 2, 3, 5, 6, 7, 10, 11, 12 and large exponents n. ...


Example 1) the first few base-3 repunit primes are 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (sequence A076481 in OEIS), corresponding to n of 3, 7, 13, 71, 103 (sequence A028491 in OEIS). The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...


Example 2) the only base-4 repunit prime is 5 (114), because 4^n-1=left(2^n+1right)left(2^n-1right), and 3 divides one of these, leaving the other as a factor of the repunit.


It is easy to prove that given n, such that n is not exactly divisible by 2 or p, there exists a repunit in base 2p that is a multiple of n.


See also

In recreational mathematics, a repdigit is a natural number composed of repeated instances of the same digit, most often in the decimal numeral system. ... A recurring decimal is an expression representing a real number in the decimal numeral system, in which after some point the same sequence of digits repeats infinitely many times. ... An all one polynomial (AOP) is a polynomial used in finite fields, specifically GF(2) (binary). ...

References

  1. ^ Harvey Dubner, New Repunit R(109297)

External links

Web sites

Books MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

  • 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.

  Results from FactBites:
 
Repunit (153 words)
In mathematics, a repunit is a number like 11, 111, or 1111 that consists of repeated units, or 1's.
A mathematical shorthand for a repunit is a capital "R" subscripted with the number of repeated units.
Prime repunits are similar to a special class of primes that remain primes after any permutation of their digits.
Repunit - Wikipedia, the free encyclopedia (317 words)
A repunit prime is a repunit that is also a prime number.
Professional mathematicians used to consider repunits an arbitrary concept, arguing that it depends on the use of decimal numerals.
The prime repunits are a subset of the permutable primes, i.e., primes that remain prime after any permutation of their digits.
  More results at FactBites »

 
 

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