q >=0.

Since RHS is odd, p and q have opposite parity.

Rewrite the equation as (p^2 - q^2 + 1)(p^2 - q^2 - 1) = 16q.

Both factors are even, so 2a(2a - 2) = 16q and a(a - 1) = 4q.

Only 1 of a and (a - 1) can be divisible by 4.

a = 4b ==> b(4b - 1) = q so 1 + 16q = (8b - 1)^2.

(a - 1) = 4b ==> (4b + 1)(b) = q so 1 + 16q = (8b + 1)^2.