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?
Proving a certain determinant $\left|\det A\right|$ is complete square
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number-theory
matrices
contest-math
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0For $6 \times 6$, the determinant is $-1$. See [here.](http://www.wolframalpha.com/input/?i=determinant+of+{{1,1,0,1,0,1},{1,0,1,0,1,0},{0,1,0,1,0,0},{1,0,1,0,0,0},{0,1,0,0,0,1},{1,0,0,0,1,0}}). Ah, but he's taking the modulus.. Anyway, some numerical data: For $n =4,5,6$, the answer is $0,1,-1$. – 2012-08-30
1 Answers
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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 .
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0I think there are too many competitions for the problems to be all original.. – 2012-08-30