A nonagonal square number is simultaneously a nonagonal number 
and a square number 
and therefore satisfies the Diophantine
equation
 |
(1)
|
Here, 
and 
are positive integers. Completing
the square and rearranging gives
 |
(2)
|
Defining 
and 
gives the Pell-like equation
 |
(3)
|
This has positive integer solutions 
, (23, 6), and (75, 20), which lead to the
family of solutions (9, 2), (23, 6), (75, 20), (247, 66), (681, 182), (2245, 600),
.... The corresponding integer solutions in 
and 
are 
,
(2, 3), (18, 33), (49, 91), (529, 989), ... (OEIS A048910
and A048911), giving the nonagonal square numbers
1, 9, 1089, 8281, 978121, 7436529, ... (OEIS A036411).
