A number which is simultaneously a nonagonal number 
and a square
number 
and therefore satisfies the Diophantine equation
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Defining 
and 
gives the Pell-like equation
 |
(3)
|
This has unit 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).
