Wieferich Primes and Monogenic Trinomials
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916037705859072 |
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| author | Jones, Lenny |
| author_facet | Jones, Lenny |
| contents | A prime $p$ is called a Wieferich prime if $2^{p-1}\equiv 1 \pmod{p^2}$.
A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N\ge 2$ is called 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 $f(θ)=0$. In this article, we show that ${\mathcal F}_p(x):=x^{2p}+2x^{p}+2$ is monogenic if and only if $p$ is not a Wieferich prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13460 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wieferich Primes and Monogenic Trinomials Jones, Lenny Number Theory A prime $p$ is called a Wieferich prime if $2^{p-1}\equiv 1 \pmod{p^2}$. A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N\ge 2$ is called 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 $f(θ)=0$. In this article, we show that ${\mathcal F}_p(x):=x^{2p}+2x^{p}+2$ is monogenic if and only if $p$ is not a Wieferich prime. |
| title | Wieferich Primes and Monogenic Trinomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.13460 |