This is equivalent to show that the following indeterminate equation is insolvable where $k=385^{97}$ or $366^{17}$

$$\left(\frac{n(n+1)}{2}\right)^2 - \left(\frac{m(m+1)}{2}\right)^2= k$$

For any integer $x$, we have $\left(\frac{x(x+1)}{2}\right)^2\equiv 0, -1\pmod{9}$. Hence the left side of the above equation modulo $9$ can only yield $0$ or $\pm 1$. But by Euler's theorem

$$385^{97}\equiv 385\times(385^{16})^6\equiv 385\equiv 7\pmod{9}$$

We can also show that for any integer $x$, $\left(\frac{x(x+1}{2}\right)^2\equiv 0, \pm 1\pmod{7}$. So the left side of the above equation modulo 7 can only be $0$, $\pm 1$, or $\pm 2$. However

$$366^{17}\equiv 2^{17}\equiv 2\times 2^4\equiv 4\pmod{7}$$