In recreational mathematics, a **repunit** is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for **rep**eated **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. ...
## Definition
The repunits are defined mathematically as Thus, the number *R*_{n} consists of *n* copies of the digit 1. The sequence of repunits starts 1, 11, 111, 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 *R*_{n} is divisible by *R*_{a}: where Φ_{d} is the *d*^{th} cyclotomic polynomial. For *p* prime, , 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 *R*_{9} is divisible by *R*_{3}—in fact, 111111111 = 111 · 1001001. The corresponding cyclotomic polynomals Φ_{3}(*x*) and Φ_{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} if *p = 2kn + 1* for some *k*. Repunit primes turn out to be rare. *R*_{n} is prime for *n* = 2, 19, 23, 317, 1031,... (sequence A004023 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.^{[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. ...
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. 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 Â± 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 (11_{4}), because , 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 2*p* 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 **^** 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. |