generalized repunit prime
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000: 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: sigma(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).

See Also: Repunit, Mersennes

Related pages (outside of this work)

References:

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