Extreme bashing

It can be shown that all integers that can be expressed as the sum of two squares can be prime factored as $2^u p_1^{a_1} \cdots p_k^{a_k} q_1^{b_1} \cdots q_r^{b_r}$ , where the $p_i$ s are of the form $4k + 1$ , the $q_j$ s are of the form $4k + 3$ , and the $b_j$ s are all even. Thus, if there exists a prime $q$ of the form $4k + 3$ in the prime factorization of a number that does not have an even power on it, i.e. $7^9$ , then it cannot be the sum of two squares.

Now, note that $204085 = 5 \times 7^4 \times 17$ . The exponent on $7$ , which is of the form $4k + 3$ , is even, so we can conclude that $\boxed{204085}$ can be written as a sum of two squares. In fact, $204085 = 441^2 + 98^2$ .

For completeness, we show that the other seven options are not possible:

$\begin{aligned} 156055 &= 5 \times 23^2 \times \mathbf{59} \\ 141336 &= 2^3 \times 3^2 \times 13 \times \mathbf{151} \\ 223293 &= 3 \times 7^4 \times \mathbf{31} \\ 123192 &= 2^3 \times 3^2 \times 29 \times \mathbf{59} \\ 139968 &= 2^6 \times \mathbf{3^7} \\ 148992 &= 2^9 \times \mathbf{3} \times 97 \\ 224939 &= \mathbf{11^3} \times 13^2. \end{aligned}$

The prime factorizations of the other seven numbers each contain a prime $p$ of the form $4k + 3$ that does not have an even power, shown in bold above. So, they cannot be the sum of two squares.