I Am A Improper Fraction. The Sum Of My Numerator And Denominator Is 1 Less Than A Perfect Square. Their Difference Is 1 More Than A Perfect Square. Their Product Is 1 Less Than A Perfect Square. Who Am I?

By trial and error, I found several fractions (a/b) that match the requirements. I believe there to be an infinite number of possible solutions.
{{a -> 1, b -> -1}, {a + b == 0, a - b == 2, a b == -1}},
{{a -> 5, b -> 3}, {a + b == 8, a - b == 2, a b == 15}},
{{a -> 13, b -> 11}, {a + b == 24, a - b == 2, a b == 143}},
{{a -> 25, b -> 23}, {a + b == 48, a - b == 2, a b == 575}},
{{a -> 41, b -> 39}, {a + b == 80, a - b == 2, a b == 1599}},
{{a -> 61, b -> 59}, {a + b == 120, a - b == 2, a b == 3599}},
{{a -> 80, b -> 63}, {a + b == 143, a - b == 17, a b == 5040}},
{{a -> 85, b -> 83}, {a + b == 168, a - b == 2, a b == 7055}},
{{a -> 113, b -> 111}, {a + b == 224, a - b == 2, a b == 12543}},
{{a -> 145, b -> 143}, {a + b == 288, a - b == 2, a b == 20735}},
{{a -> 181, b -> 179}, {a + b == 360, a - b == 2, a b == 32399}},
{{a -> 221, b -> 219}, {a + b == 440, a - b == 2, a b == 48399}}

The solution set includes  1/-1, 5/3, 13/11, 25/23, 41/39, 61/59, 80/63, 85/83, 113/111, 145/143, 181/179, 221/219.