Download PDF - Finding integers k for which a given Diophantine equation has no solution in kth powers of integersAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Finding integers k for which a given Diophantine equation has no solution in kth powers of integers

Authors 1, 2

Affiliations

  1. School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540, U.S.A.
  2. Department of Mathematics, University of Georgia, Athens, Georgia 30602, U.S.A.

Bibliography

  1. [AHB] L. M. Adleman and D. R. Heath-Brown, The first case of Fermat's last theorem, Invent. Math. 79 (1985), 409-416.
  2. [An] N. C. Ankeny, The insolubility of sets of Diophantine equations in the rational numbers, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 880-884.
  3. [AE] N. C. Ankeny and P. Erdős, The insolubility of classes of Diophantine equations, Amer. J. Math. 76 (1954), 488-496.
  4. [BM] W. D. Brownawell and D. W. Masser, Vanishing sums in function fields, Math. Proc. Cambridge Philos. Soc. 100 (1986), 427-434.
  5. [Ch] V. Chvátal, Linear Programming, Freeman, New York 1983.
  6. [CJ] J. H. Conway and A. J. Jones, Trigonometric diophantine equations (On vanishing sums of roots of unity), Acta Arith. 30 (1976), 229-240.
  7. [DL] H. Davenport and D. J. Lewis, Homogeneous additive equations, Proc. Royal Soc. Ser. A 274 (1963), 443-460.
  8. [Fa] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349-366, Erratum, Proc. Royal Soc. Ser. A 75 (1984), 381.
  9. [Fo] E. Fouvry, Théorème de Brun-Titchmarsh; application au théorème de Fermat, Proc. Royal Soc. Ser. A 79 (1985), 383-407.
  10. [G1] A. Granville, Diophantine equations with varying exponents (with special reference to Fermat's Last Theorem), Doctoral thesis, Queen's University, Kingston, Ontario, 1987, 209 pp.
  11. [G2] A. Granville, Some conjectures in Analytic Number Theory and their connection with Fermat's Last Theorem, in: Analytic Number Theory, B. C. Brendt, H. G. Diamond, H. Halberstam, A. Hildebrand (eds.) Birkhäuser, Boston 1990, 311-326.
  12. [G3] A. Granville, The set of exponents for which Fermat's Last Theorem is true, has density one, C. R. Math. Acad. Sci. Canada 7 (1985), 55-60.
  13. [HB] D. R. Heath-Brown, Fermat's Last Theorem for ``almost all'' exponents, Bull. London Math. Soc. 17 (1985), 15-16.
  14. [L] H. W. Lenstra,Jr., Vanishing sums of roots of unity, in: Proc. Bicentennial Cong. Wiskundig Genootschap, Vrije Univ., Amsterdam 1978, 249-268.
  15. [M] H. B. Mann, On linear relations between roots of unity, Mathematika 12 (1965), 107-117.
  16. [NS] D. J. Newman and M. Slater, Waring's problem for the ring of polynomials, J. Number Theory 11 (1979), 477-487.
  17. [Ri] P. Ribenboim, An extension of Sophie Germain's method to a wide class of diophantine equations, J. Reine Angew. Math. 356 (1985), 49-66.
  18. [V] H. S. Vandiver, On classes of Diophantine equations of higher degrees which have no solutions, Proc. Nat. Acad. Sci., U.S.A. 32 (1946), 101-106.
Acta Arithmetica
1991-1992/60/3
Export to PBN, Opens in new tab
Pages:
203-212
Main language of publication
English
Received
1990-03-23
Accepted
1991-03-29
Published
1992
Exact and natural sciences