On the first case of Fermat's last theorem
type: Item
instance of: scholarly article
title: On the first case of Fermat's last theorem
zbMATH Open document ID: 0571.10016
DOI: 10.1016/0022-314X(85)90033-2
author: Francisco Thaine
published in: Journal of Number Theory
publication date: 1985-01-01T00:00:00Z
review text: Suppose that there are integers a,b,c prime to each other and to p (an odd prime) such that \(a^ p+b^ p+c^ p=0\). Let \(u=-a/b\). Then, by a classical result, (*) \(B_ n\sum^{p-1}_{k=1}u^ k k^{-n}\equiv 0\) (mod p), where the \(B_ n\) are Bernoulli numbers \((n=0,...,p-2).\) The author gives a simple new expression and a simple proof for certain congruences derived from (*) by D. Mirimanoff. These congruences imply the famous conditions \(2^{p-1}\equiv 1\) and \(3^{p-1}\equiv 1\) (mod \(p^ 2)\). He also proves the following complement to the congruences (*): provided that the ''real'' factor \(h^+_ p\) of the class number of the p-th cyclotomic field is prime to p, if p \(| B_ n\) with n even, \(2\leq n\leq p-3\), then p divides \(\sum^{p-1}_{k=1}k^{n-1} u^ k\).
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 3910478
zbMATH Keywords: Kummer-Mirimanoff congruences
zbMATH Keywords: first case of Fermat's last theorem
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: class number of the p-th cyclotomic field
MaRDI profile type: Publication
cites work: Q5521595
cites work: Q4126428
cites work: Q3904661
cites work: Fermat's Last Theorem: Its History and the Nature of the Known Results Concerning It
cites work: Polynomials generalizing binomial coefficients and their application to the study of Fermat's last theorem
full work available at URL: 0022-314x(85)90033-2
OpenAlex ID: W1973593276
description: scientific article; zbMATH DE number 3910478