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: (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:

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