# generalized repunit prime

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: (bn-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 blength n base blength 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).

Related pages (outside of this work)

References:

AG1974
I. O. Angell and H. J. Godwin, "Some factorizations of 10n± 1," Math. Comp., 28 (1974) 307--308.  MR 48:8366
Beiler1964
A. Beiler, Recreations in the theory of numbers, Dover Pub., New York, NY, 1964.
BLSTW88
J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman and S. S. Wagstaff, Jr., Factorizations of bn ± 1, b=2,3,5,6,7,10,12 up to high powers, Amer. Math. Soc., 1988.  Providence RI, pp. xcvi+236, ISBN 0-8218-5078-4. MR 90d:11009 (Annotation available)
CD95
C. Caldwell and H. Dubner, "The near repunit primes 1n-k-1011k," J. Recreational Math., 27 (1995) 35--41.
Dubner2002
Dubner, Harvey, "Repunit R49081 is a probable prime," Math. Comp., 71:238 (2002) 833--835 (electronic).  (http://dx.doi.org/10.1090/S0025-5718-01-01319-9) MR 1885632 (Abstract available)
Dubner93
H. Dubner, "Generalized repunit primes," Math. Comp., 61 (1993) 927--930.  MR 94a:11009
Jaroma2009
Jaroma, John H., On primes and pseudoprimes in the generalized repunitsCongr. Numer., In "Proceedings of the Fortieth Southeastern International Conference on Combinatorics, Graph Theory and Computing," Vol, 195, 2009.  pp. 105--114, MR 2584289
Oblath1956
R. Obláth, "Une propriété des puissances parfaites," Mathesis, 65 (1956) 356--364.
Rotkiewicz1987
A. Rotkiewicz, "Note on the diophantine equation 1 + x + x2 + ... + xn = ym," Elem. Math., 42:3 (1987) 76.  MR 88c:11020
Salas2011
Salas, Christian, "Base-3 repunit primes and the Cantor set," Gen. Math., 19:2 (2011) 103--107.  MR 2818401
WD86
H. C. Williams and H. Dubner, "The primality of R1031," Math. Comp., 47:176 (1986) 703--711.  MR 87k:11141
Yates82
S. Yates, Repunits and repetends, Star Publishing Co., Inc., Boynton Beach, Florida, 1982.  pp. vi+215, MR 83k:10014