Discover

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.

Contents

Definition

Repunit primes

Generalizations

See also

References

External links

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).

Repunit primes

Historically, 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=rac{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 subsituted 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'' 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.^[1] He later announced there are no others from ''R''₈₆₄₅₃ to ''R''₂₀₀₀₀₀.^[2] On July 15 2007 Maksym Voznyy announced ''R''₂₇₀₃₄₃ to be probably prime ^[3], along with his intent to search to 400000.
It has been conjectured that there are infinitely many repunit primes,[1] 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 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

★ Repdigit

★ Recurring decimal

★ All one polynomial - Another generalization

References

1. Harvey Dubner, ''New Repunit R(109297)''
2. Harvey Dubner, ''Repunit search limit''
3. Maksym Voznyy, ''New PRP Repunit R(270343)''

External links

Web sites

★

★ The main tables of the Cunningham project.

★ Repunit at The Prime Pages by Chris Caldwell.

★ Repunits and their prime factors at 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.