Suppose the positive integers satisfy

………. (1)

Or

is a Triangular number [ ]

numbers of the form are Second pentagonal numbers.

Equation (1) gives us the Pell equation:

The first few solutions are:

(x, y) = (0,0), (22,18), (2180,1780), (213642,174438), (20934760,17093160), (2051392862,1674955258)

Note that

Show that for all that satisfy

are all perfect squares.

Summarizing results:

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