A number which is simultaneously a heptagonal number 
and pentagonal 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, 5), (17, 13), (53, 41), (133, 103), .... The
integer solutions in 
and 
are then given by 
, (42, 54), (2585, 3337), (160210, 206830), (9930417,
12820113) ... (OEIS A046198 and A046199),
corresponding to the heptagonal pentagonal numbers 1, 4347, 16701685, 64167869935,
246532939589097, ... (OEIS A048900).
