A number which is simultaneously a heptagonal number 
and square 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 
, (13, 4), and (57, 18). Additional solutions can be obtained from the unit Pell equation, and correspond to integer solutions when 
, (6, 9), (49, 77), (961, 1519), ... (OEIS A046195 and A046196), corresponding to the heptagonal square numbers 1, 81, 5929, 2307361, 168662169, 12328771225, ... (OEIS A036354).
