The Prime Glossary: generalized repunit

generalized repunit
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A repunit is a number whose expansion (in base 10) is a string of ones (for example: 11 and 11111111). A generalized repunit (base b) is one whose expansion base b is all ones. For example, the Mersenne numbers are the generalized repunits in base 2. Here is a formula for the n "digit" generalized repunit (base b):

(bⁿ-1)/(b-1).

In the special case that b is prime, the generalized repunit is the value of the sum of divisors function:

(b^n-1).

We can also generalize the notion of a repunit prime: a generalized repunit prime is a generalized repunit that is prime. For example, the generalized repunit primes with less than 100 decimal digits are as follows.

base b length n base b length n

2 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127 (n > 337 ) 19 19, 31, 47, 59, 61 (n > 83)

3 3, 7, 13, 71, 103 (any others have n > 211) 17 3, 5, 7, 11, 47, 71 (n > 83)

4 2 (no others) 18 2 (n > 83)

5 3, 7, 11, 13, 47, 127,149 (n > 149) 20 3, 11, 17, (n > 79)

6 2, 3, 7, 29, 71, 127, (n > 131) 21 3, 11, 17, 43 (n > 79)

7 5, 13 (n > 127) 22 2, 5, 79 (n > 79)

8 3 (no others) 23 5 (n > 79)

9 (none) 24 3, 5, 19, 53, 71 (n > 79)

10 2, 19, 23 (n > 101 ) 25 (none)

11 17, 19, 73 (n > 97) 26 7, 43 (n > 73)

12 2, 3, 5, 19, 97 (n > 97) 27 3 (no others)

13 5, 7 (n > 97) 28 2, 5, 17 (n > 71)

14 3, 7, 19, 31, 41 (n > 89) 29 5 (n > 71)

15 3, 43, 73 (n > 89) 30 2, 5, 11 (n > 71)

16 2 (no others)

(You might want to explain why the lists for b=4, 8, 9, 16, 25 and 27 are so short.)

A couple of larger examples include: (1956¹⁸⁰¹-1)/1955 (5925 decimal digits), (218⁹⁷¹-1)/217 (2269 decimal digits) and (3⁴¹⁷⁷-1)/2 (1993 decimal digits).

See Also: Repunit, Mersennes

Related pages (outside of this work)

All the generalized repunit primes on the list of the 5000 largest known primes

References:

CD95
C. Caldwell and H. Dubner, "The near repunit primes 1_n-k-1011_k," J. Recreational Math., 27 (1995) 35--41.
Dubner93
H. Dubner, "Generalized repunit primes," Math. Comp., 61 (1993) 927--930. MR 94a:11009