Prove that the product of any two numbers between two consecutive squares is never a perfect square In essence, I want to prove that the product of any two distinct elements in the set $\{n^2, n^2+1, ... , (n+1)^2-1\}$ is never a perfect square for a positive integers $n$. I have no idea on how to prove it, but I've also yet to find a counterexample to this statement. Can anyone help?

For any two numbers $n^2+a,\ n^2+b;\ 0<a<b<(2n+1)$, their product will satisfy $n^4<n^4+(a+b)n^2+ab<(n^2+2n+1)^2$. 
All of the squares between $n^4$ and $(n^2+2n+1)^2$ will have the form $(n^2+m)^2=n^4+2mn^2+m^2;\ 1\le m\le 2n$ 
If $n^4+(a+b)n^2+ab$ is a perfect square, it will be one of the squares between $n^4$ and $(n^2+2n+1)^2$ and hence equal to $n^4+2mn^2+m^2$ for some $m$.
Thus $(a+b)=2m;\ ab=m^2$ for some $m$
Rearranging, we get $m^2=\frac{a^2+2ab+b^2}{4}=ab=\frac{4ab}{4}$, or $a^2+b^2=2ab$.
This implies both $a\mid b$ and $b\mid a$, meaning $b=a$ and the numbers being multiplied to obtain a perfect square are not distinct.
Added by edit: John Omielan comments (for the specific case $k=1$) that my original answer fails to consider possible solutions of the form $a+b=2m-k;\ ab=m^2+kn^2$. He separately provides a more complete answer that addresses those cases. Marty Cohen comments that I can only properly conclude $n^4+(a+b)n^2+ab=n^4+2mn^2+m^2 \Rightarrow n^2(a+b-2m)=(m^2-ab)$. Let me address those shortcomings.
If $(n^2+a)(n^2+b)=(n^2+m)(n^2+m)$ then either $a=b=m$ (addressed in my original answer) or $a<m<b$ (which this edit will address). If $n^2(a+b-2m)=(m^2-ab)$, then $(m^2-ab)$ is divisible by $n^2$, so $(m^2-ab)=kn^2=n^2(a+b-2m)$, or $(a+b-2m)=k\Rightarrow a+b=2m+k$. 
$m,a,b$ have limits on their sizes. $m<b\le 2n\Rightarrow m^2< 4n^2$. Also $a<b\le2n \Rightarrow ab<4n^2$ Hence, $|(m^2-ab)|<4n^2 \Rightarrow |k|=(a+b-2m)<4$. For $(a+b)=2m+k,\ |k|=0,1,2,3$, where $k=0$ corresponds to the case where $a=b=m$.
The midpoint $t$ between $a$ and $b$ is the average $t=\frac{a+b}{2}$. Let $b-t=r,\ t-a=r$. Note that if $a$ and $b$ have different parity, $t$ and $r$ may have half integral values. Finally, $b-a=2r$, but since $b\le2n,\ a\ge 1$, then $2r=b-a<2n\Rightarrow r<n$.
$(n^2+a)(n^2+b)=((n^2+t)-r)((n^2+t)+r)=(n^2+t)^2-r^2$. Therefore, unless $m>t$, $(n^2+a)(n^2+b)<(n^2+m)^2$. But $t=\frac{a+b}{2}=m-\frac{k}{2}$. $m-t=\frac{1}{2},1,\frac{3}{2}$. Within the constraints of the original question, $m$ must be larger than the average of $a$ and $b$, but it must be very close to that average.
Substituting $t-\frac{k}{2}$ for $m$ in $(n^2+m)^2$, we get $((n^2+t)-\frac{k}{2})^2$. Can that equal $(n^2+a)(n^2+b)=((n^2+t)-r)((n^2+t)+r)=(n^2+t)^2-r^2$? Letting $s=(n^2+t)$ for simplicity in keeping track during expansion, we ask whether $(s-\frac{k}{2})^2=(s-r)(s+r)=s^2-r^2\Rightarrow ks-\frac{k^2}{4}=r^2\Rightarrow k(n^2+t)-\frac{k^2}{4}=r^2$?
$r<n\Rightarrow r^2<n^2$. It is obvious for $3\ge k>1$ that $k(n^2+t)-\frac{k^2}{4}>n^2>r^2$. For $k=1$, $n^2+t-\frac{1}{4}>n^2 \iff t>\frac{1}{4}$. For $a<m<b$, $\min(a+b)=4 \Rightarrow \min(t)=2; 2>\frac{1}{4}$.
There are no solutions to the question for $a<m<b$.
