A number which is simultaneously octagonal and heptagonal. Let 
denote the 
th octagonal number and
the 
th
heptagonal number, then a number which is both
octagonal and hexagonal satisfies the equation 
, or
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Therefore, defining
 |  |  |
(3)
|
 |  |  |
(4)
|
gives the second-order Diophantine equation
 |
(5)
|
The first few solutions are 
, (7, 4), (73, 40), (157, 86), .... These give the
integer solutions (1, 1), (345, 315), (166145, 151669), ... (OEIS A048904
and A048905), corresponding to the octagonal
heptagonal numbers 1, 297045, 69010153345, ... (OEIS A048906).
