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Date: 26-5-2020
1414
Date: 2-11-2020
717
Date: 11-2-2020
751
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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."
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