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