On the p-adic zeros of polynomials with Bernoulli coefficients
type: Item
instance of: scholarly article
title: On the p-adic zeros of polynomials with Bernoulli coefficients
zbMATH Open document ID: 0563.10013
DOI: 10.1007/BF01194893
author: Willem Louw Fouché
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: If p is an odd prime, let \(\beta_ p(X)=\sum_{k}(B_ k/k)(X^{p-1- k}-1)\), where \(B_ k\) is the kth Bernoulli number and the summation extends over \(1\leq k\leq p-2\). In this paper we prove the following theorem: Let p be an odd prime number and let \(\alpha\) be an integer such that \(\alpha\geq 2\). If \(p>\alpha^{\pi (\alpha)} \log \alpha\), where \(\pi\) (\(\alpha)\) denotes the number of primes \(\leq \alpha\), and \(\alpha^{p-1}\equiv 1 (mod p^ 2)\), then, for some positive constant C(\(\alpha)\), the polynomial \(\beta_ p(X)\) has \(n>C(\alpha) \log \log p\) distinct simple zeros \(x_ i\) in the ring of p-adic integers. Furthermore, for some integral sequence \(0\leq \tau (1)<...<\tau (n)\leq n\), it follows that \(x_ i\equiv \alpha^{\tau (i)} (mod p)\) for all \(1\leq i\leq n\).
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11S05
zbMATH DE Number: 3897022
zbMATH Keywords: Fermat quotients
zbMATH Keywords: Bernoulli number
zbMATH Keywords: ring of p-adic integers
MaRDI profile type: Publication
cites work: A Reciprocity Law for Polynomials with Bernoulli Coefficients
cites work: Groups of homotopy spheres. I
cites work: Units and class groups in number theory and algebraic geometry
cites work: Q5670687
full work available at URL: https://doi.org/10.1007/bf01194893
OpenAlex ID: W2154925009
description: scientific article; zbMATH DE number 3897022