###CONJECTURE

Let $A= (c_0,c_1,\ldots,c_n)$ be a circulant matrix, i.e if $(c_0,c_1,\ldots,c_n)$ is the first column of $A$ then the $i$^th column of $A$ is obtained by applying the permutation $(1,2,..,n)^{i-1}$.

Assume $A \in GL_n(Z)$, i.e $A$ with integer entries and determinant $\pm 1$ and moreover $c_0+c_1+\ldots+c_n=\pm 1$.

Then there exists one $j$ such that $c_j=\pm 1$ and $c_i=0$ for all $i$ different from $j$.

Is this conjecture true?

What if we add the assumption that $n=p$ a prime?

Thanks for any idea! Fabienne

CONJECTURE

Let $A= (c_0,c_1,\ldots,c_n)$ be a circulant matrix, i.e if $(c_0,c_1,\ldots,c_n)$ is the first column of $A$ then the $i$^th column of $A$ is obtained by applying the permutation $(1,2,..,n)^{i-1}$.

Assume $A \in GL_n(Z)$, i.e $A$ with integer entries and determinant $\pm 1$ and moreover $c_0+c_1+\ldots+c_n=\pm 1$.

Then there exists one $j$ such that $c_j=\pm 1$ and $c_i=0$ for all $i$ different from $j$.

Is this conjecture true?

What if we add the assumption that $n=p$ a prime?

Thanks for any idea! Fabienne

###CONJECTURE

Let $A= (c0,c1,..,cn)$ be a circulant matrix, i.e if $(c0,c1,..,cn)$ is the first column of $A$ then the $i$^th column of $A$ is obtained by applying the permutation $(1,2,..,n)^{n-1}$.

Assume $A$ in $GL_n(Z)$, i.e $A$ with integer entries and determinant +/-1 and moreover $c0+c1+..+cn=+-1$.

Then there exists one $j$ such that $cj=+/-1$ and $ci=0$ for all $i$ different from $j$.

Is this conjecture true?

What if we add the assumption that $n=p$ a prime?

Thanks for any idea! Fabienne

###CONJECTURE

Let $A= (c_0,c_1,\ldots,c_n)$ be a circulant matrix, i.e if $(c_0,c_1,\ldots,c_n)$ is the first column of $A$ then the $i$^th column of $A$ is obtained by applying the permutation $(1,2,..,n)^{i-1}$.

Assume $A \in GL_n(Z)$, i.e $A$ with integer entries and determinant $\pm 1$ and moreover $c_0+c_1+\ldots+c_n=\pm 1$.

Then there exists one $j$ such that $c_j=\pm 1$ and $c_i=0$ for all $i$ different from $j$.

Is this conjecture true?

What if we add the assumption that $n=p$ a prime?

Thanks for any idea! Fabienne

CONJECTURE: Let A= (c0,c1,..,cn) be a circulant matrix, i.e if (c0,c1,..,cn) is the first column of A then the i-th column of A is obtained by applying the permutation (1,2,..,n)^{n-1}. Assume A in GL_n(Z), i.e A with integer entries and determinant=+-1 and moreover c0+c1+..+cn=+-1. Then there exists one j such that cj=+-1 and ci=0 for all i diffrent from j.

IS IT TRUE?

What if we add the assumption that n=p a prime? Thanks for any idea! Fabienne

###CONJECTURE

Let $A= (c0,c1,..,cn)$ be a circulant matrix, i.e if $(c0,c1,..,cn)$ is the first column of $A$ then the $i$^th column of $A$ is obtained by applying the permutation $(1,2,..,n)^{n-1}$. 

Assume $A$ in $GL_n(Z)$, i.e $A$ with integer entries and determinant +/-1 and moreover $c0+c1+..+cn=+-1$.

Then there exists one $j$ such that $cj=+/-1$ and $ci=0$ for all $i$ different from $j$.

Is this conjecture true?

What if we add the assumption that $n=p$ a prime?

Thanks for any idea! Fabienne