Find the smallest solution in positive integers of $x^{2} - 14 y^{2} = 1$ .

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Solution

$x^{2} - 1 = 14 y^{2}$
Now checking by putting y =1,2,3... untill we get solution
$y = 1; => x^{2} = 15$ not possible
$y = 2; => x^{2} = 57$ not possible
$y = 3; => x^{2} = 127$ not possible
$y = 4; => x^{2} = 225 => x = 15$
Hence, smallest solution is $(x, y) = (15, 4)$ .

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