Generalized Wieferich primes and monogenic trinomials

Let $b\ge 2$ be an integer and let $p\ge 3$ be a prime. We say that $p$ is a {\em generalized Wieferich prime base $b$}, or more succinctly, a {\em base-$b$ Wieferich prime,} if $b^{p-1}\equiv 1 \pmod{p^2}$. When $b=2$, $p$ is also known simply as a Wieferich prime. Let $f(x)\in {\mathbb Z}[x]$ be a monic polynomial of degree $N\ge 2$. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. Recently, the third author proved that $x^{2p}+2x^p+2$ is monogenic if and only if $p$ is not a Wieferich prime. In this article, we generalize this result to $x^{2n}+bx^n+b$ with certain restrictions on $b\ge 2$ and $n\ge 3$.