A number which is simultaneously a nonagonal number 
and pentagonal 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 solutions in 
corresponding to solutions which are integral in 
and 
of 
,
(14, 21), (7189, 10981), (165026, 252081), (86968201, 132846121), ... (OEIS A048913
and A048914), giving the nonagonal pentagonal
numbers 1, 651, 180868051, 95317119801, 26472137730696901, ... (OEIS A048915).
