True or False

First Solution:

$\large N=\underbrace{11\dots11}_{1997}\times10^{1999} + \underbrace{22\dots22}_{1998}\times 10 + 5$

$=\large\frac{1}{9}(10^{1997} - 1)\times 10^{1999} + \frac{2}{9}(10^{1998} - 1)\times 10 + 5$

$=\large\frac{1}{9} ( 10^{3996} + 2\times5\times10^{1998}+25) = (\frac{1}{3}(10^{1998}+5))^2$

$=(\large\frac{1\overbrace{00\dots00}^{1997}5}{3})^2 =(\underbrace{33\dots33}_{1997}5)^2$

Second Solution:

$\large 9N = 1\underbrace{00\dots00}_{1996}1\underbrace{00\dots00}_{1997}25 = 10^{3996} + 10^{1999} + 25 = (10^{1998} + 5)^2 \implies N \text{ is a square. }$