Problem 7

If $n=0, 10^{n}-1=0 \implies 3x=3y=0$

If $n=1$ , trial and error gives $(x, y)=(1, 0), (0, 1)$ .

If $n> 1$ : $10^{n}$ is a multiple of $2^{2}=4$ Hence right hand side is congruent to $3 mod 4$ . However perfect squares (check 0, 1, 2, 3) can only be congruent to $0, 1 mod 4$ . Summing two perfect squares can only give $0+0=0, 0+1=1, (1+0=1), 1+1=2$ modulo 4 and this gives a contradiction.

Hence there are three solutions $(x, y, n)=(0,0,0), (0,1,1), (1,0,1)$ .