Editing Repunit (section)

=== Algebra factorization of generalized repunit numbers ===
If ''b'' is a [[perfect power]] (can be written as ''m''<sup>''n''</sup>, with ''m'', ''n'' integers, ''n'' > 1) differs from 1, then there is at most one repunit in base-''b''. If ''n'' is a [[prime power]] (can be written as ''p''<sup>''r''</sup>, with ''p'' prime, ''r'' integer, ''p'', ''r'' >0), then all repunit in base-''b'' are not prime aside from ''R<sub>p</sub>'' and ''R<sub>2</sub>''. ''R<sub>p</sub>'' can be either prime or composite, the former examples, ''b'' = −216, −128, 4, 8, 16, 27, 36, 100, 128, 256, etc., the latter examples, ''b'' = −243, −125, −64, −32, −27, −8, 9, 25, 32, 49, 81, 121, 125, 144, 169, 196, 216, 225, 243, 289, etc., and ''R<sub>2</sub>'' can be prime (when ''p'' differs from 2) only if ''b'' is negative, a power of −2, for example, ''b'' = −8, −32, −128, −8192, etc., in fact, the ''R<sub>2</sub>'' can also be composite, for example, ''b'' = −512, −2048, −32768, etc. If ''n'' is not a prime power, then no base-''b'' repunit prime exists, for example, ''b'' = 64, 729 (with ''n'' = 6), ''b'' = 1024 (with ''n'' = 10), and ''b'' = −1 or 0 (with ''n'' any natural number). Another special situation is ''b'' = −4''k''<sup>4</sup>, with ''k'' positive integer, which has the [[aurifeuillean factorization]], for example, ''b'' = −4 (with ''k'' = 1, then ''R<sub>2</sub>'' and ''R<sub>3</sub>'' are primes), and ''b'' = −64, −324, −1024, −2500, −5184, ... (with ''k'' = 2, 3, 4, 5, 6, ...), then no base-''b'' repunit prime exists. It is also conjectured that when ''b'' is neither a perfect power nor −4''k''<sup>4</sup> with ''k'' positive integer, then there are infinity many base-''b'' repunit primes.