A number which is simultaneously a heptagonal number 
and triangular 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 basic solutions 
, (7, 3), and (18, 8). Additional solutions can be
obtained from the unit Pell equation, and correspond
to integer solutions when 
, (5, 10), (221, 493), (1513, 3382), ... (OEIS A046193 and A039835),
corresponding to the heptagonal triangular numbers 1, 55, 121771, 5720653, 12625478965,
... (OEIS A046194).
