AI 中文总结
该研究将关于x^(2p)+2x^p+2单基因性与Wieferich素数关联的结论，推广到特定条件下的多项式x^(2n)+bx^n+b，拓展了单基因三项式的判定相关成果。
AI 中文摘要
设整数b≥2，素数p≥3，若b^(p−1)≡1 mod p²，则称p为基b的广义Wieferich素数，简称基b Wieferich素数；当b=2时，p即为通常的Wieferich素数。设f(x)∈ℤ[x]是次数N≥2的首一多项式，若f(x)在ℚ上不可约，且{1,θ,θ²,…,θ^(N−1)}是ℚ(θ)的整数环的一组基（其中f(θ)=0），则称f(x)是单基因的。近期第三作者证明了x^(2p)+2x^p+2为单基因当且仅当p不是Wieferich素数。本文将该结果推广到满足b≥2、n≥3及特定限制条件的多项式x^(2n)+bx^n+b。
英文摘要
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 $\{1,θ,θ^2,\ldots,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. 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 $n\ge 3$.
