The n−th triangular number is given by . how can one find all integers n for which the sum of the (n − 1)−st, the n−th,
and the (n + 1)−st triangular numbers is a square?
Google told me to look in the papers:
Triangular Numbers and Perfect Squares
Tom Beldon and Tony Gardiner
The Mathematical Gazette
Vol. 86, No. 507 (Nov., 2002), pp. 423-431
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88.53 Comments on 'Triangular Numbers and Perfect Squares'
Michael D. Hirschhorn
The Mathematical Gazette
Year, volume, issue, and the page range: Vol. 88, No. 513 (Nov., 2004), pp. 500-503
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We have
Now multiply by and add and subtract 9 to get
Let and multiply both sides by to get
By letting we see we have which is almost Pell's Equation.
Luckily though this is solved similarly. I won't solve it, but here for reference on how to solve it (look for the part about solving ).