TITLE: Proving a certain determinant $\left|\det A\right|$ is complete square
QUESTION [11 upvotes]: Consider the following matrix
$$
A_{ij}=
\begin{cases}
1\quad\text{ if }\space (i+j)\space\text{ is prime,}\\
0\quad\text{ otherwise.}
\end{cases}
$$
How can one prove that $\left|\det A\right|$ is a complete square?

REPLY [5 votes]: It was a problem posed at an annual contest for undergraduates called IMC (International Mathematics Contest). The official site of the site http://www.imc-math.org . It was the fifth problem of the second day (year 2008), which means it is pretty hard to solve. Here is a reference of the official site http://www.imc-math.org.uk/imc2008/day2_solutions.pdf . Also there is a discussion here at artofproblemsolving forum http://www.artofproblemsolving.com/Forum/viewtopic.php?f=79&t=217339 .