Repunit
A repunit is a number consisting of copies of the single digit 1. The term "repunit" was coined by Beiler (1966), who also gave the first tabulation of known factors.
In base-10, repunits have the form
 |  |  |
(1)
|
 |  |  |
(2)
|
Repunits 
therefore have exactly 
decimal digits.
Amazingly, the squares of the repunits 
give the Demlo numbers, 
, 
, 
, ...
(OEIS A002275 and A002477).
The number of factors for the base-10 repunits for 
, 2, ... are
1, 1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, ... (OEIS A046053).
The base-10 repunit probable primes 
occur for 
, 19, 23, 317, and 1031, 49081, 86453, 109297, and 270343
(OEIS A004023; Madachy 1979, Williams and Dubner
1986, Ball and Coxeter 1987, Granlund, Dubner 1999, Baxter 2000), where 
is the
largest proven prime (Williams and Dubner 1986). T. Granlund completed
a search up to 
in 1998 using two months of CPU
time on a parallel computer. The search was extended by Dubner (1999), culminating
in the discovery of the probable prime 
. A number
of larger repunit probable primes have since been
found, as summarized in the following table.
 | discoverer(s) | date |
| 49081 | H. Dubner (1999, 2002) | Sep. 9, 1999 |
| 86453 | L. Baxter (2000) | Oct. 26, 2000 |
| 109297 | P. Bourdelais (2007), H. Dubner (2007) | Mar. 26-28, 2007 |
| 270343 | M. Voznyy and A. Budnyy (2007) | Jul. 11, 2007 |
Every prime repunit is a circular prime.
Repunit can be generalized to base 
, giving a base-
repunit as number of the form
 |
(3)
|
This gives the special cases summarized in the following table.
 |  | name |
| 2 |  | Mersenne number  |
| 10 |  | repunit  |
The idea of repunits can also be extended to negative bases. Except for requiring 
to be odd, the math is very similar (Dubner and
Granlund 2000).
 | OEIS |  -repunits |
 | A066443 | 1, 7, 61, 547, 4921, 44287, 398581, ... |
 | A007583 | 1, 3, 11, 43, 171, 683, 2731, ... |
| 2 | A000225 | 1, 3, 7, 15, 31, 63, 127, ... |
| 3 | A003462 | 1, 4, 13, 40, 121, 364, ... |
| 4 | A002450 | 1, 5, 21, 85, 341, 1365, ... |
| 5 | A003463 | 1, 6, 31, 156, 781, 3906, ... |
| 6 | A003464 | 1, 7, 43, 259, 1555, 9331, ... |
| 7 | A023000 | 1, 8, 57, 400, 2801, 19608, ... |
| 8 | A023001 | 1, 9, 73, 585, 4681, 37449, ... |
| 9 | A002452 | 1, 10, 91, 820, 7381, 66430, ... |
| 10 | A002275 | 1, 11, 111, 1111, 11111, ... |
| 11 | A016123 | 1, 12, 133, 1464, 16105, 177156, ... |
| 12 | A016125 | 1, 13, 157, 1885, 22621, 271453, ... |
Williams and Seah (1979) factored generalized repunits for 
and
. A (base-10) repunit can be prime
only if 
is prime, since
otherwise 
is a binomial
number which can be factored algebraically. In fact, if 
is even,
then 
. As with positive bases,
all the exponents of prime repunits with negative bases are also prime.
 | OEIS |  of prime  -repunits |
 | A057178 | 5, 11, 109, 193, 1483, ... |
 | A057177 | 5, 7, 179, 229, 439, 557, 6113, ... |
 | A001562 | 5, 7, 19, 31, 53, 67, 293, ... |
 | A057175 | 3, 59, 223, 547, 773, 1009, 1823, ... |
 | A057173 | 3, 17, 23, 29, 47, 61, 1619, ... |
 | A057172 | 3, 11, 31, 43, 47, 59, 107, ... |
 | A057171 | 5, 67, 101, 103, 229, 347, 4013, ... |
 | A007658 | 3, 5, 7, 13, 23, 43, 281, ... |
 | A000978 | 3, 5, 7, 11, 13, 17, 19, ... |
| 2 | A000043 | 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, ... |
| 3 | A028491 | 3, 7, 13, 71, 103, 541, 1091, 1367, ... |
| 5 | A004061 | 3, 7, 11, 13, 47, 127, 149, 181, 619, ... |
| 6 | A004062 | 2, 3, 7, 29, 71, 127, 271, 509, 1049, ... |
| 7 | A004063 | 5, 13, 131, 149, 1699, ... |
| 10 | A004023 | 2, 19, 23, 317, 1031, ... |
| 11 | A005808 | 17, 19, 73, 139, 907, 1907, 2029, 4801, ... |
| 12 | A004064 | 2, 3, 5, 19, 97, 109, 317, 353, 701, ... |
Yates (1982) published all the repunit factors for 
, a portion
of which are reproduced in the Wolfram
Language notebook by Weisstein. Brillhart et al. (1988) gave a table of
repunit factors which cannot be obtained algebraically, and a continuously updated
version of this table is now maintained online. These tables include factors for
(with 
odd) and 
(with 
even and odd).
After algebraically factoring 
, these types of factors are sufficient
for complete factorizations.
The sequence of least 
such that 
is
prime for 
, 2, ... are 2, 3, 2, 3, 2, 5, 3, 0,
2, 17, 2, 5, ... (OEIS A084740), and the sequence
of least 
such that 
is
prime for 
, 2, ... are 3, 3, 3, 5, 3, 3, 0, 3,
5, 5, 5, 3, ... (OEIS A084742).
A Smith number can be constructed from every factored repunit.
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is a Probable Prime." Math.
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."
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." §A3 in Unsolved
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1994.
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." Math.
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. Math.
Comput. 33, 1337-1342, 1979.
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