Suppose $n_2$ denotes the binary representation of the integer number $n$. Let $X_2(n)=[1_22_2\ldots n_2]$,$n\geq2$, be a binary vector which is obtained by concatenating of binary representation of the numbers from $1$ to $n$. Also, let $X_2^m(n)$,$0\leq m\leq n-1$,denotes the cyclically $m$ shift of the entries of the vector $X_2(n)$. For example, we have $$X_2^0(n)=X_2(n)$$ $$X_2^1(n)=[n_21_22_2\ldots(n-1)_2]$$ $$X_2^2(n)=[(n-1)_2n_21_22_2\ldots(n-2)_2]$$ and so on.

For two binary vectors $X$ and $Y$ (with same length), suppose $|X\cap Y|$ denotes the number of ones common to both $X$ and $Y$.

The conjecture is:

for all $m$ and $k$, we have $|X_2^m(n)\cap X_2^k(n)|\cong 0 \mod 2$ if and only if $n=2^s-1$, for some integer number $s$.

The conjecture is tested for many integer numbers.