##
**Some historical aspects of the Fermat problem.**
*(Czech)*
Zbl 0831.11025

The purpose of this paper is to make the mathematical public acquainted with the short history and some results on the Fermat problem. Because of the enormous greatness of this area, the choice of the results must be very subjective. The methods of arithmetical algebraic geometry, particularly Wiles’ proof of Fermat’s Last Theorem, are mentioned only very shortly in this article, but the reader is referred to the paper of J. Nekovář [Math. Bohem. 119, 79-96 (1994; Zbl 0813.11016)]for a more extensive review.

Section 1 is devoted to the personality of Fermat himself, some of his characteristic properties and some of his results from number theory. Particularly his invention “the method of infinite descent” is described in great detail. Section 2 deals with Kummer’s discovery of ideal theory (the theory of ideal numbers), Kummer’s definition of regular and irregular primes and with Kummer’s theorem on the validity of Fermat’s conjecture for regular prime exponents.

Section 3 contains the notion of Bernoulli numbers and their connection to the Fermat problem. In particular, M. Eichler’s criterion [Acta Arith. 11, 129-131 (1965; Zbl 0135.094)]transferred to the irregularity index of a prime and M. Krasner’s criterion [C. R. Acad Sci., Paris 199, 256-258 (1934; Zbl 0010.00702)]using the “strong” irregularity index of a prime are mentioned.

The final part of this paper (Section 4) includes some results on the Fermat quotients concerning Fermat’s conjecture starting with the celebrated theorem of Wieferich (1909) and ending with the latest result by A. Granville and M. Monagan [Trans. Am. Math. Soc. 306, 329-359 (1988; Zbl 0645.10018)]. In the conclusion recent criteria of the first case of Fermat’s Last Theorem for an odd prime \(\ell\) concerning sums \(\sum_{\ell k/N< x<\ell (k+1)} {1\over x}\) are quoted.

Section 1 is devoted to the personality of Fermat himself, some of his characteristic properties and some of his results from number theory. Particularly his invention “the method of infinite descent” is described in great detail. Section 2 deals with Kummer’s discovery of ideal theory (the theory of ideal numbers), Kummer’s definition of regular and irregular primes and with Kummer’s theorem on the validity of Fermat’s conjecture for regular prime exponents.

Section 3 contains the notion of Bernoulli numbers and their connection to the Fermat problem. In particular, M. Eichler’s criterion [Acta Arith. 11, 129-131 (1965; Zbl 0135.094)]transferred to the irregularity index of a prime and M. Krasner’s criterion [C. R. Acad Sci., Paris 199, 256-258 (1934; Zbl 0010.00702)]using the “strong” irregularity index of a prime are mentioned.

The final part of this paper (Section 4) includes some results on the Fermat quotients concerning Fermat’s conjecture starting with the celebrated theorem of Wieferich (1909) and ending with the latest result by A. Granville and M. Monagan [Trans. Am. Math. Soc. 306, 329-359 (1988; Zbl 0645.10018)]. In the conclusion recent criteria of the first case of Fermat’s Last Theorem for an odd prime \(\ell\) concerning sums \(\sum_{\ell k/N< x<\ell (k+1)} {1\over x}\) are quoted.

Reviewer: L.Skula (Brno)

### MSC:

11D41 | Higher degree equations; Fermat’s equation |

11-03 | History of number theory |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

01A55 | History of mathematics in the 19th century |

01A60 | History of mathematics in the 20th century |