This is a modification of [an unanswered problem on the math StackExchange](https://math.stackexchange.com/questions/614238/integers-n-such-that-ii1i2-cdots-in-is-real-or-pure-imaginary/619597#619597).


When is the product $(1+1)(1+4)…(1+n^2)$ a perfect square? 


If $(1+1)(1+4)…(1+n^2)=k^2$ then one possibility is $n=3$, $k=10$. Could there be other integer solutions for $(n,k)$?


Surely the answer is NO, but I am unable to prove that $n=3$, $k=10$ is the only possibility.



When is the product $(1+1)(1+4)…(1+n^2)$ a perfect square?
[Javier Cilleruelo](https://mathoverflow.net/users/31020/javier) has shown that $n=3$ is the only solution: see <http://www.uam.es/personal_pdi/ciencias/cillerue/Papers/squares-sinlogo.pdf> . 


A couple more comments: apparently Chebyshev already showed that the largest prime factor of $\prod\_{j=1}^{n} (1+j^2)$ is bigger than $Cn$ for any constant $C$ provided that $n$ is large. This would solve the problem for all large $n$ (taking $C=2$ in Chebyshev's work). The problem of finding the largest prime factor of $\prod\_{j=1}^{n} f(j)$ for a polynomial $f$ has been studied by many authors (Erdos, Hooley, Heath-Brown ...); for a first improvement of Chebyshev's work see Erdos's paper <http://www.renyi.hu/~p_erdos/1952-07.pdf> .


