Using Euler's criterion we can tell if an integer is a quadratic residue modulo a prime. For example, if the prime was 11, then we could test the integers from 1 to 10 and determine that 1, 3, 4, 5, and 9 are quadratic residues in this case. If we instead found the quadratic residues by starting at 1 and squaring successive integers...
$1^2 \equiv 1$ (mod 11)
$2^2 \equiv 4$ (mod 11)
$3^2 \equiv 9$ (mod 11)
$4^2 \equiv 5$ (mod 11)
$5^2 \equiv 3$ (mod 11)
$6^2 \equiv 3$ (mod 11)
$7^2 \equiv 5$ (mod 11)
$8^2 \equiv 9$ (mod 11)
$9^2 \equiv 4$ (mod 11)
$10^2 \equiv 1$ (mod 11)
... we get the same answer but in the order: 1, 4, 9, 5, 3.
How can this order be predicted?