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?
I agree that that us the idea. Also,
1 < 7/4 < 8/1 < 9
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.