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 $444$.

Now we show that all such numbers are so expressed.

In Squares Ending in Repeated Digits, we have shown the only numbers with squares ending in $44$ ends in:
 * $12, 38, 62, 88$

hence any number with square ending in $444$ must also end in those numbers.

Suppose $\sqbrk {axy}^2 \equiv 444 \pmod {1000}$, where $a < 10$ and $\sqbrk {xy}$ is in the above list.

For $\sqbrk {xy} = 12$:

This is a contradiction.

Similarly for $\sqbrk {xy} = 88$:

Again, a contradiction.

For $\sqbrk {xy} = 38$:

The solutions to $600 a \equiv 0 \pmod {1000}$ are $a = 0$ or $5$.

Hence:
 * $\paren {500 n + 38}^2 \equiv 444 \pmod {1000}$

Similarly, for $\sqbrk {xy} = 62$:

The solutions to $400 a + 400 \equiv 0 \pmod {1000}$ are $a = 4$ or $9$.

Hence:
 * $\paren {500 n + 462}^2 \equiv \paren {500 n - 38}^2 \equiv 444 \pmod {1000}$