Nonagonal Square Number

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).

REFERENCES:

Sloane, N. J. A. Sequences A048910, A048911, and A036411 in "The On-Line Encyclopedia of Integer Sequences."