TITLE: Determine all integers $i$ such that $(i-29)(i+29)$ is a square number
QUESTION [5 upvotes]: Determine all integers $i$ such that 
  $$(i-29)(i+29)$$ 
  is a square number.

I’ve tried some substitutions but none of them worked...
I think that the only solutions are $i=\pm 29$, but I still don’t know how to prove it.

REPLY [13 votes]: Hint: $$i^2-29^2=j^2\implies (i-j)(i+j)= 29^2$$
$$
\begin{array}{c|c}
i+j & i-j & 2i& i \\
\hline
1& 29^2 & 1+29^2 &421\\
29&29&58&29\\
29^2&1& 1+29^2& 421\\
-1& -29^2 & -1-29^2 &-421\\
-29&-29&-58&-29\\
-29^2&-1& -1-29^2&-421
\end{array}
$$