TITLE: Is the number $333{,}333{,}333{,}333{,}333{,}333{,}333{,}333{,}334$ a perfect square?
QUESTION [20 upvotes]: I know that if the number is a perfect square then it will be congruent to $0$ or $1$ (mod $4$).  Now since the number is even, I know that it is either $0$ or $2$ (mod $4$).  How would I go about answering this?

REPLY [5 votes]: More generally. "Find all positive integers $k,n$ such that $\frac{(6n-2)^k-1}{3}+1$ is a square."
Answer. Clearly such a square, if it exists, has to be odd. Suppose then that there exists a positive integer $x$ such that
$$\frac{1}{3}((6n-2)^k-1)+1=(2x+1)^2 \implies 12x(x+1)=(6n-2)^k-1.$$
As far as one between $x$ and $x+1$ has to be even, then $\upsilon_2((6n-2)^k-1) \ge 3$, which is clearly impossible since it's a odd number. $\blacksquare$