Notes on polynomials $(1+X)^n + (-1)^n(X^n+1)$ concerning the regularity problem for symmetric power sums in 3 variables

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 $6$.