Given 2 real numbers a & b, b>a.

Find four distinct integers p,q,r, and s, complying with:

p^2 + q^2 = r^2 + s^2

a < p/q < r/s < b

(In reply to

Is this the idea? by broll)

I agree that that us the idea. Also,

1 < 7/4 < 8/1 < 9

where

s=1 r=8, q=4, p=7, a = 1, b=9.

The challenge, of course, is to come up with a method for finding a satisfying p,q,r,s for any a and b, and particularly for a and b arbitrarily close. Or at least, to prove that these always exist.