Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A new criterion for the first case of Fermat's last theorem


Authors: Karl Dilcher and Ladislav Skula
Journal: Math. Comp. 64 (1995), 363-392
MSC: Primary 11D41; Secondary 11Y50
DOI: https://doi.org/10.1090/S0025-5718-1995-1248969-6
MathSciNet review: 1248969
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that if the first case of Fermat's last theorem fails for an odd prime l, then the sums of reciprocals modulo l, $ s(k,N) = \sum 1/j\;(kl/N < j < (k + 1)l/N)$ are congruent to zero $ \bmod\;l$ for all integers N and k with $ 1 \leq N \leq 46$ and $ 0 \leq k \leq N - 1$. This is equivalent to $ {B_{l - 1}}(k/N) - {B_{l - 1}} \equiv 0 \pmod l$, where $ {B_n}$ and $ {B_n}(x)$ are the nth Bernoulli number and polynomial, respectively. The work can be considered as a result on Kummer's system of congruences.


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

  • [1] T. Agoh, On the Kummer-Mirimanoff congruences, Acta Arith. 60 (1990), 141-156. MR 1061635 (91d:11020)
  • [2] G. Almkvist and A. Meurman, Values of Bernoulli polynomials and Hurwitz's zeta function at rational points, C. R. Math. Rep. Acad. Sci. Canada 13 (1991), 104-108. MR 1112244 (92g:11023)
  • [3] K. Bartz and J. Rutkowski, On the von Staudt-Clausen theorem, C. R. Math. Rep. Acad. Sci. Canada 15 (1993), 46-48. MR 1214216 (94b:11017)
  • [4] P. T. Bateman, G. B. Purdy, and S. S. Wagstaff, Jr., Some numerical results on Fekete polynomials, Math. Comp. 29 (1975), 7-23. MR 0480293 (58:468)
  • [5] Z. I. Borevich and I. R. Shafarevich, Number theory, Academic Press, New York, 1966. MR 0195803 (33:4001)
  • [6] J. Brillhart, J. Tonascia, and P. Weinberger, On the Fermat quotient, Computers in Number Theory, Academic Press, London and New York, 1971, pp. 213-222. MR 0314736 (47:3288)
  • [7] P. Cikánek, A special extension of Wieferich's criterion, Math. Comp. 62 (1994), 923-930. MR 1216257 (94g:11023)
  • [8] D. Coppersmith, Fermat's last theorem (case 1) and the Wieferich criterion, Math. Comp. 54 (1990), 895-902. MR 1010598 (90h:11024)
  • [9] L. E. Dickson, History of the theory of numbers, Vol. 1, Divisibility and Primality, Chelsea, New York, 1962.
  • [10] G. Eisenstein, Eine neue Gattung zahlentheoretischer Funktionen welche von zwei Elementen abhängen und durch gewisse lineare Funktional-Gleichungen definiert werden, Bericht über die zur Bekanntmachung geeigneten Verhandlungen der Königl. Preuss. Akademie der Wissenschaften zu Berlin (1850), 36-42. (See also Mathematische Werke, 2nd ed., Gotthold Eisenstein, Band II, Chelsea, New York, 1989, pp. 705-711.)
  • [11] J. W. L. Glaisher, On the residues of $ {r^{p - 1}}$ to modulus $ {p^2},{p^3}$ etc., Quart. J. Math. 32 (1901), 1-27.
  • [12] A. Granville and M. B. Monagan, The first case of Fermat's last theorem is true for all prime exponents up to 714,591,416,091,389, Trans. Amer. Math. Soc. 306 (1988), 329-359. MR 927694 (89g:11025)
  • [13] A. Granville, The Kummer-Wieferich-Skula approach to the first case of Fermat's Last Theorem, Advances in Number Theory (F. Q. Gouvêa and N. Yui, eds.), (Proc. Third Conference of the Canadian Number Theory Assoc., August 18-24, 1991, Queen's University at Kingston), Clarendon Press, Oxford, 1993, pp. 479-497. MR 1368443 (96m:11020)
  • [14] N. G. Gunderson, Derivation of criteria for the first case of Fermat's last theorem and the combination of these criteria to produce a new lower bound for the exponent, Thesis, Cornell University, 1948.
  • [15] E. E. Kummer, Einige Sätze über die aus den Wurzeln der Gleichung $ {\alpha ^\lambda } = 1$ gebildeten complexen Zahlen für den Fall, dass die Klassenanzahl durch $ \lambda $ theilbar ist, nebst Anwendung derselben auf einen weiteren Beweis des letzten Fermat'schen Lehrsatzes, Math. Abh. Königl. Akad. Wiss. zu Berlin (1857), pp. 41-74. (Collected Papers, Vol. I, Springer-Verlag, Berlin and New York, 1975, pp. 639-692.)
  • [16] E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math. (2) 39 (1938), 350-360. MR 1503412
  • [17] M. Lerch, Zur Theorie des Fermatschen Quotienten $ ({a^{p - 1}} - 1)/p = q(a)$, Math. Ann. 60 (1905), 471-490. MR 1511321
  • [18] D. Mirimanoff, Sur le dernier théorème de Fermat, C. R. Acad. Sci. Paris 150 (1910), 204-206.
  • [19] P. L. Montgomery, New solutions of $ {a^{p - 1}} \equiv 1\;\pmod {p^2}$, Math. Comp. 61 (1993), 361-363. MR 1182246 (94d:11003)
  • [20] F. Pollaczek, Über den grossen Fermat'schen Satz, Akad. Wiss. Wien, Abt. IIa 126 (1917), 45-49.
  • [21] P. Ribenboim, 13 lectures on Fermat's Last Theorem, Springer-Verlag, New York, 1979. MR 551363 (81f:10023)
  • [22] -, The book of prime number records, Springer-Verlag, New York, 1988. MR 931080 (89e:11052)
  • [23] L. Skula, On the Kummer's system of congruences, Comment. Math. Univ. St. Paul. 35 (1986), 137-163. MR 864734 (87m:11016)
  • [24] -, Some consequences of the Kummer system of congruences, Comment. Math. Univ. St. Paul. 39 (1990), 19-40. MR 1059524 (91i:11030)
  • [25] -, Fermat's last theorem and the Fermat quotients, Comment. Math. Univ. St. Paul. 41 (1992), 35-54. MR 1166223 (93f:11028)
  • [26] Zhi-Hong Sun and Zhi-Wei Sun, Fibonacci numbers and Fermat's Last Theorem, Acta Arith. 60 (1992), 371-388. MR 1159353 (93e:11025)
  • [27] J. J. Sylvester, Sur une propriété des nombres premiers qui se rattache au théorème de Fermat, C. R. Acad. Sci. Paris 52 (1861), 161-163; also in The Collected Mathematical Papers, Vol. II, Chelsea, New York, 1973, pp. 229-231.
  • [28] J. W. Tanner and S. S. Wagstaff, Jr., New bound for the first case of Fermat's last theorem, Math. Comp. 53 (1989), 743-750. MR 982371 (90h:11028)
  • [29] H. S. Vandiver, Extension of the criteria of Wieferich and Mirimanoff in connection with Fermat's last theorem, J. Reine Angew. Math. 144 (1914), 314-318.
  • [30] I. M. Vinogradov, Elements of number theory, Dover, New York, 1954. MR 0062138 (15:933e)
  • [31] L. Washington, Introduction to cyclotomic fields, Springer-Verlag, New York, 1982. MR 718674 (85g:11001)
  • [32] A. Wieferich, Zum letzten Fermat'schen Theorem, J. Reine Angew. Math. 136 (1909), 293-302.
  • [33] A. Granville and Zhi-Wei Sun, Values of Bernoulli polynomials, Pacific J. Math. (to appear). MR 1379289 (98b:11018)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11D41, 11Y50

Retrieve articles in all journals with MSC: 11D41, 11Y50


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1995-1248969-6
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society