A number which is simultaneously a heptagonal number 
and hexagonal number 
. Such numbers exist when
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Substituting 
and 
gives the Pell-like quadratic Diophantine equation
 |
(3)
|
which has solutions 
, (7, 3), (18, 8), (47, 21), (123, 55), .... The
integer solutions in 
and 
are then given by 
, (221, 247), (71065, 79453), (22882613, 25583539),
... (OEIS A048902 and A048901),
corresponding to the heptagonal hexagonal numbers 1, 121771, 12625478965, 1309034909945503,
... (OEIS A048903).
