Integers whose Squares end in 444

Theorem
The sequence of positive integers whose square ends in $444$ begins:
 * $38, 462, 538, 962, 1038, 1462, 1538, 1962, 2038, 2462, 2538, 2962, 3038, 3462, \ldots$

Proof
All such $n$ are of the form $500 m + 38$ or $500 m - 38$:

and it is seen that all such numbers end in $444$.