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Date: 5-10-2020
564
Date: 30-1-2021
2397
Date: 8-2-2020
574
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A number which is simultaneously a nonagonal number and a triangular 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 , (9, 3), and (19, 7), which lead to the family of solutions (5, 1), (9, 3), (19, 7), (61, 23), (135, 51), (299, 113), (971, 367), .... The corresponding integer solutions in and are , (10, 25), (154, 406), (2449, 6478), ... (OEIS A048907 and A048908), giving the nonagonal triangular numbers 1, 325, 82621, 20985481, 5330229625, 1353857339341, ... (OEIS A048909).
REFERENCES:
Sloane, N. J. A. Sequences A048907, A048908, and A048909 in "The On-Line Encyclopedia of Integer Sequences."
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