Editing Mersenne prime (section)

====Repunit primes====
{{main|Repunit}}
The other way to deal with the fact that {{math|''b''<sup>''n''</sup> − 1}} is always divisible by {{math|''b'' − 1}}, it is to simply take out this factor and ask which values of {{math|''n''}} make
:<math>\frac{b^n-1}{b-1}</math>
be prime. (The integer {{math|''b''}} can be either positive or negative.) If, for example, we take {{math|''b'' {{=}} 10}}, we get {{math|''n''}} values of:
:2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... {{OEIS|id=A004023}},<br>corresponding to primes 11, 1111111111111111111, 11111111111111111111111, ... {{OEIS|id=A004022}}.
These primes are called repunit primes. Another example is when we take {{math|''b'' {{=}} −12}}, we get {{math|''n''}} values of:
:2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... {{OEIS|id=A057178}},<br>corresponding to primes −11, 19141, 57154490053, ....
It is a conjecture that for every integer {{math|''b''}} which is not a [[perfect power]], there are infinitely many values of {{math|''n''}} such that {{math|{{sfrac|''b''<sup>''n''</sup> − 1|''b'' − 1}}}} is prime. (When {{math|''b''}} is a perfect power, it can be shown that there is at most one {{math|''n''}} value such that {{math|{{sfrac|''b''<sup>''n''</sup> − 1|''b'' − 1}}}} is prime)

Least {{math|''n''}} such that {{math|{{sfrac|''b''<sup>''n''</sup> − 1|''b'' − 1}}}} is prime are (starting with {{math|''b'' {{=}} 2}}, {{math|0}} if no such {{math|''n''}} exists)
:2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, ... {{OEIS|id=A084740}}

For negative bases {{math|''b''}}, they are (starting with {{math|''b'' {{=}} −2}}, {{math|0}} if no such {{math|''n''}} exists)
:3, 2, 2, 5, 2, 3, 2, 3, 5, 5, 2, 3, 2, 3, 3, 7, 2, 17, 2, 3, 3, 11, 2, 3, 11, 0, 3, 7, 2, 109, 2, 5, 3, 11, 31, 5, 2, 3, 53, 17, 2, 5, 2, 103, 7, 5, 2, 7, 1153, 3, 7, 21943, 2, 3, 37, 53, 3, 17, 2, 7, 2, 3, 0, 19, 7, 3, 2, 11, 3, 5, 2, ... {{OEIS|id=A084742}} (notice this OEIS sequence does not allow {{math|''n'' {{=}} 2}})

Least base {{math|''b''}} such that {{math|{{sfrac|''b''<sup>prime(''n'')</sup> − 1|''b'' − 1}}}} is prime are
:2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, ... {{OEIS|id=A066180}}

For negative bases {{math|''b''}}, they are
:3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, ... {{OEIS|id=A103795}}