A number which is simultaneously octagonal and pentagonal. Let 
denote the 
th octagonal number and
the 
th
pentagonal number, then a number which is both
octagonal and pentagonal satisfies the equation 
, or
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Therefore, defining
 |  |  |
(3)
|
 |  |  |
(4)
|
gives the Pell equation
 |
(5)
|
The first few solutions are 
, (5, 4), (11, 8), (31, 22), (65, 46), .... These
give the solutions 
, (1, 1), (2, 5/3), (16/3, 4), (11, 8), ...,
of which the integer solutions are (1, 1), (11, 8), (1025, 725), (12507, 8844), ...
(OEIS A046187 and A046188),
corresponding to the octagonal pentagonal numbers 1, 176, 1575425, 234631320, 2098015778145,
... (OEIS A046189).
