Can you find integer solutions of

This can be viewed as a Pell Equation

in integers, so we know that there exist integers and such that:

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There are an infinite number of solutions to that. There are infinite sets when n=1 and n=4 and n=9, it seems to miss 16, back at 25 and 36. There is an interesting relationship at n=36, the value of “a” is defined by the pairs of 18t+-5, which goes {{5}, {13, 23}, {31,41},{49,59},{67,77},{85,95},…}

Here are some values at n=36, format is {n, k, a}

{36,10,5}

{36,738,13}

{36,7598,23}

{36,25334,31}

{36,77934,41}

{36,159334,49}

{36,335434,59}

{36,558258,67}

{36,974498,77}

{36,1447610,85}

…

{36,288372401010,1795}..t=100

{36,294852439010,1805}..t=100

The last two were found by solving back for k

Paul.

I seem to miss at k = 8