_n_ = 2 = (11)
_n_ = 6 = 111 x 1,001
_n_ = 10 = 11,111 x 100,001
_n_ = 14 = 1,111,111 x 10,000,001
_n_ = 18 = 111,111,111 x 1,000,000,001
In the above two tables _n_ is of the form 4_m_ + 2. When _n_ is of the
form 4_m_ the factors may be written down as follows:--
_n_= 4 = (11) x (101)
_n_ = 8 = (11) x (101) x 10,001
_n_ = 12 = (11) x (101) x 100,010,001
_n_ = 16 = (11) x (101) x 1,000,100,010,001.
When _n_ = 2, we have the prime number 11; when _n_ = 3, the factors are
3 . 37; when _n_ = 6, they are 11 . 3 . 37 . 7. 13; when _n_ = 9, they
are 3^2 . 37 . 333,667. Therefore we know that factors of _n_ = 18 are
11. 3^2 . 37 . 7 . 13 . 333,667, while the remaining factor is composite
and can be split into 19 . 52579. This will show how the working may be
simplified when _n_ is not prime.