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