Fermat’s Last Theorem and the Fermat quotients. (English) Zbl 0753.11016

Kummer formulated a criterion for the solvability of the Fermat equation \((*)\;x^ \ell+y^ \ell=z^ \ell\) in Case I (\(\ell\) an odd prime not dividing \(xyz\)) in terms of a system of congruences mod \(\ell\) involving Bernoulli numbers and certain polynomials, later called Mirimanoff polynomials. The author [ibid. 35, 137–163 (1986; Zbl 0604.10006)] introduced another system of congruences which in a certain sense is equivalent to Kummer’s system. The present paper continues the author’s study of his system. Special congruences are obtained which enable one to prove the vanishing mod \(\ell\) of some Fermat quotients \((p^{\ell-1}- 1)/\ell\) and of related power sums (here \(p\) is a prime). As an application it is shown that if \((*)\) is solvable in Case I, then the sums \(\sum x^{\ell-2}\) \((k\ell/N<x<(k+1)\ell/N)\) vanish mod \(\ell\) for all \(N\in\{2,\ldots,10\}\cup\{12\}\) and all \(k=0,\ldots,N-1\).


11D41 Higher degree equations; Fermat’s equation
11A15 Power residues, reciprocity
11A07 Congruences; primitive roots; residue systems


Zbl 0604.10006