On the first case of Fermat's last theorem
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 \(\sum^{p-1}_{k=1}k^{n-1} u^ k\).
class number of the p-th cyclotomic field
scientific article; zbMATH DE number 3910478