Generalized Wieferich primes and monogenic polynomials
Abstract
Let b, p∈ ℤ with b≥ 2 and p≥ 3 a prime. If bp−1≡ 1 pmodp2, then p is called a {\em generalized Wieferich prime base b}, or more succinctly, a {\em base-b Wieferich prime}. When b=2, p is also known simply as a Wieferich prime. We say that a monic polynomial f(x)∈ ℤ[x] is {\em monogenic} if f(x) is irreducible over ℚ and {1,θ,θ2,…,θ°(f)−1} is a basis for the ring of integers of ℚ(θ), where f(θ)=0. Recently, necessary and sufficient conditions for the monogenicity of the trinomials x2n+bxn+b were given that included base-b Wieferich prime congruences, and also congruences involving a certain Lucas sequence. In this article, we prove a similar result for a different class of trinomials that does not rely on any congruence condition involving a Lucas sequence. Furthermore, we extend this result to a related class of N-nomials, for any N≥ 4.
Related
- Perfect numbers, Wieferich primes and the solutions of binom2nn≡ 2n bmod n
- Radical defects, Wieferich primes, and the abc conjecture
- Necessary and sufficient condition for reciprocal polynomials to be monogenic
- Higher order Teodorescu transform in the theory of generalized partial-slice monogenic functions
- Primes and the Möbius function over quadratic polynomials
BibTeX
@misc{jones2026generalized,
title = {{Generalized Wieferich primes and monogenic polynomials}},
author = {Lenny Jones},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13904},
eprint = {2609.13904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13904},
}
RIS
TY - UNPB
TI - Generalized Wieferich primes and monogenic polynomials
AU - Lenny Jones
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13904
UR - https://arxiv.org/abs/2609.13904
AB - Let b, p∈ ℤ with b≥ 2 and p≥ 3 a prime. If b^p−1≡ 1 pmodp^2, then p is called a {\em generalized Wieferich prime base b}, or more succinctly, a {\em base-b Wieferich prime}. When b=2, p is also known simply as a Wieferich prime. We say that a monic polynomial f(x)∈ ℤ[x] is {\em monogenic} if f(x) is irreducible over ℚ and {1,θ,θ^2,…,θ^°(f)−1} is a basis for the ring of integers of ℚ(θ), where f(θ)=0. Recently, necessary and sufficient conditions for the monogenicity of the trinomials x^2n+bx^n+b were given that included base-b Wieferich prime congruences, and also congruences involving a certain Lucas sequence. In this article, we prove a similar result for a different class of trinomials that does not rely on any congruence condition involving a Lucas sequence. Furthermore, we extend this result to a related class of N-nomials, for any N≥ 4.
ER -