(1) Suppose

(p, q)

is a lattice point on the parabola (that is

p

and

q

are integers), then

q = {ap}^{2}

and for any integer value of

k

it follows that

k^{2} q = a (kp)^{2}

so the point

(kp, k^{2} q)

is another lattice point on the parabola. As

k

can take infinitely many values there are infinitely many lattice points on the parabola.

(2)The hyperbola has equation:

$(x - y) (x + y) = 84 .$

Noting that $(x - y)$ and $(x + y)$ have the same parity (both odd or both even) and that their product is even we see that both $(x - y)$ and $(x + y)$ must be even. Take $x - y = 2 u$ and $x + y = 2 v$ then

$uv = \frac{84}{4} = 21$

and so the only possibilities with $u, v \geq 0$ are:

$(u, v) = (1, 21), (3, 7), (7, 3), (21, 1) .$

Now $x = u + v$ and $y = v - u$ giving the lattice points:

$(22, 20), (10, 4), (10, - 4), (22, - 20)$

and (by symmetry):

$(- 22, 20), (- 10, 4), (- 22, - 20), (- 10, - 4)$