Repunit
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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.
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Definition
The repunits are defined mathematically as
- <math>R_n={10^n-1\over9}\qquad\mbox{for }n\ge1.<math>
Thus, the number Rn consists of n copies of the digit 1. The sequence of repunits starts 1, 11, 111, 1111,... (sequence A002275 in OEIS).
Repunit primes
Historically, the definition of repunits was motivated by recreational mathematicians looking for prime factors of such numbers. Wikipedia contains a list of repunit factorizations.
It is easy to show that if n is divisible by a, then Rn is divisible by Ra. For example, 9 is divisible by 3, and indeed R9 is divisible by R3—in fact, 111111111 = 111 · 1001001. 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.
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. It has been conjectured that there are infinitely many repunit primes (http://primes.utm.edu/glossary/page.php?sort=Repunit).
Generalizations
Professional mathematicians used to consider repunits an arbitrary concept, arguing that it depends on the use of decimal numerals. But the arbitrariness can be removed by generalizing the idea to base-b repunits:
- <math>R_n^{(b)}={b^n-1\over b-1}\qquad\mbox{for }n\ge1.<math>
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.
The prime repunits are a subset of the permutable primes, i.e., primes that remain prime after any permutation of their digits.
See also
External links
Web sites
- Repunit (http://mathworld.wolfram.com/Repunit.html) article on MathWorld (http://mathworld.wolfram.com).
- The main tables (http://www.cerias.purdue.edu/homes/ssw/cun/third/pmain901) of the Cunningham project (http://www.cerias.purdue.edu/homes/ssw/cun/).
- Repunit (http://primes.utm.edu/glossary/page.php?sort=Repunit) at The Prime Pages (http://primes.utm.edu/) by Chris Caldwell.
- Repunits and their prime factors (http://www.worldofnumbers.com/repunits.htm) at World!Of Numbers (http://www.worldofnumbers.com).
Books
- A. Beiler, Recreations in the theory of numbers. ISBN 0486210960. Chapter 11, of course.
- Paulo Ribenboim, The New Book Of Prime Number Records. ISBN 0387944575.it:Repunit