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Three summation criteria for Fermat's last theorem


Author: H. Schwindt
Journal: Math. Comp. 40 (1983), 715-716
MSC: Primary 10-04; Secondary 10B15
DOI: https://doi.org/10.1090/S0025-5718-1983-0689484-X
MathSciNet review: 689484
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper extends the search for solutions of the congruences

$\displaystyle \sum\limits_1^{[p/6]} {\frac{1}{i} \equiv 0,} \quad \sum\limits_1... ...ext{and}}\quad \sum\limits_{[p/6] + 1}^{[p/5]} {\frac{1}{i} \equiv 0\;\pmod p} $

to the limit $ p < 600000$. The only solutions found were $ p = 61$ in the first case, in the second $ p = 205129$, and in the third case $ p = 109$ and $ p = 491$.

References [Enhancements On Off] (What's this?)

  • [1] P. Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer-Verlag, Berlin and New York, 1979, pp. 155-164. MR 551363 (81f:10023)
  • [2] H. S. Vandiver "A new type of criteria for the first case of Fermat's Last Theorem," Ann. of Math., v. 26, 1925, pp. 88-94.
  • [H] Schwindt, "Eine Bemerkung zu einem Kriterium von H. S. Vandiver," Jahresbericht d. Deutschen Math. Verein, v. 43, 1933-34, pp. 229-232.
  • [3] Emma Lehmer, "On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson," Ann. of Math., v. 39, 1938, pp. 350-359. MR 1503412
  • [4] D. E. Knuth, The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, Addison-Wesley, Reading, Mass., 1973, p. 325. MR 633878 (83i:68003)
  • [5] D. H. Lehmer, "On Fermat's quotient, base two," Math. Comp., v. 36, 1981, pp. 289-290. MR 595064 (82e:10004)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1983-0689484-X
Article copyright: © Copyright 1983 American Mathematical Society

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