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Refining the conditions on the Fermat quotient

  • Andrew J. Granville (a1)
Abstract

The aim of this paper is to establish a result on a family of congruences arising from the Fermat quotient: this result has an interesting application to the Fermat problem. For over 150 years the Fermat problem has been divided into two cases; the First Case being the assertion that for each odd prime p,

has no solution in non-zero integers x, y, z. Kummer's work on ideal numbers led, in 1847, to the complete solution of the Fermat problem for regular primes.

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[1] L. M. Adleman and D. R. Heath-Brown . The first case of Fermat's last theorem. Invent. Math. 79 (1985), 409416.

[2] E. Fouvry . Théorème de Brun-Titchmarsh; application au théorème de Fermat. Invent. Math. 79 (1985), 383407.

[4] D. H. Lehmer . On Fermat's quotient base two. Math. Comput. 36, 153 (1981), 289290.

[6] P. Ribenboim . Thirteen lectures on Fermat's last theorem (Springer-Verlag, 1979).

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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