  Let $K$ be a field and $f _{n}(X) = (X + 1) ^{n} + (-1) ^{n}(X ^{n} + 1) \in
K[X]$, for each $n \in \mathbb N$. This note shows that the polynomials $f
_{m}(X)$ and $f _{m'}(X)$ are relatively prime, for some distinct indices $m$
and $m ^{\prime}$ at most equal to $100$, if and only if the product $mm
^{\prime }$ is divisible by $6$.
