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The orders of solutions of the Kummer system of congruences


Author: Ladislav Skula
Journal: Trans. Amer. Math. Soc. 343 (1994), 587-607
MSC: Primary 11D41; Secondary 11B68
DOI: https://doi.org/10.1090/S0002-9947-1994-1196218-5
MathSciNet review: 1196218
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Abstract: A new method concerning solutions of the Kummer system of congruences (K) (modulo an odd prime l) is developed. This method is based on the notion of the Stickelberger ideal. By means of this method a new proof of Pollaczek's and Morishima's assertion on solutions of (K) of orders 3, 6 and 4 $ \bmod\; l$ is given. It is also shown that in case there is a solution of $ (K)\, \not\equiv \, 0, \pm 1\;\pmod l$, then for the index of irregularity $ i(l)$ of the prime l we have $ i(l) \geq [\sqrt[3]{{l/2}}]$.


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  • [1] Z. J. Borevich and J. R. Shafarevich, Number theory, Academic Press, London and New York, 1966. MR 0195803 (33:4001)
  • [2] H. Brückner, Explizites Reciprozitätsgesetz und Anwendungen, Vorlesungen Fachbereich Math. Univ. Essen, Heft 2, 1979, 83 pp..
  • [3] J. Buhler, R. Crendall, and B. Sompolski, Irregular primes to one million, Math. Comp. 59 (1992), 717-722. MR 1134717 (93a:11106)
  • [4] L. Carlitz, A generalization of Maillet's determinant and a bound for the first factor of the class number, Proc. Amer. Math. Soc. 12 (1961), 256-261. MR 0121354 (22:12093)
  • [5] M. Eichler, Eine Bemerkung zur Fermatschen Vermutung, Acta Arith. 11 (1965), 129-131, 261. MR 0174523 (30:4724)
  • [6] 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. Akademie der Wissenschaften zu Berlin, 1850, pp. 36-452 (p. 41). Math. Werke, Gotthold Eisenstein, Band II, Chelsea, New York, 2nd ed., 1989, pp. 705-711 (p. 710).
  • [7] R. Ernvall, Generalized Bernoulli numbers, generalized irregular primes, and class number, Ann. Univ. Turku Ser. A Math. 178 (1979), 72 pp. MR 533377 (80m:12002)
  • [8] -, An upper bound for the index of $ \chi $-irregularity, Mathematika 32 (1985), 39-44. MR 817105 (87e:11024)
  • [9] 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)
  • [10] 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 Univ., 1948.
  • [11] H. Kleboth, Untersuchung über Klassenzahl und Reziprozitätsgesetz im Körper der 6l-ten Einheitswurzeln und die Diophantische Gleichung $ {x^{21}} + {3^l}{y^{21}} = {z^{21}}$ für eine Primzahl l grosser als 3, Dissertation, Universität Zürich, 1955, 37 pp. MR 0098075 (20:4537)
  • [12] E. E. Kummer, Einige Sätze über die aus den Wurzeln der Gleichung $ {\alpha ^\lambda } = 1$ gebildeten comlexen Zahlen, für den Fall, dass die Klassenanzahl durch $ \lambda $ theilbar ist, nebst Anwendung derselben auf einen weiteren Beweis des letzten Fermat'schen Lehrsatzes, Math. Abhandl. Königl. Akad. Wiss. zu Berlin, 1857, pp. 41-74. (Collected Papers, I, 639-692).
  • [13] E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math. 39 (1938), 350-360. MR 1503412
  • [14] H. W. Leopoldt, Eine Verallgemeinerung der Bernoullischen Zahlen, Abh. Math. Sem. Univ. Hamburg 22 (1958), 131-140. MR 0092812 (19:1161e)
  • [15] R. Lidl and H. Niederreiter, Finite fields, Addison-Wesley, London, 1983. MR 746963 (86c:11106)
  • [16] T. Morishima, Über die Fermatsche Quotienten, Japan J. Math. 8 (1931), 159-173.
  • [17] F. Pollaczek, Über den grossen Fermat 'schen Satz, Sitzungsber. Akad. Wiss. Wien Abt. Ha 126 (1917), 45-59.
  • [18] P. Ribenboim, 13 lectures on Fermat's Last Theorem, Springer-Verlag, New York, Heidelberg, and Berlin, 1979. MR 551363 (81f:10023)
  • [19] L. Skula, Non-possiblity to prove infinity of regular primes from some theorems, J. Reine Angew. Math. 291 (1977), 162-181. MR 0447098 (56:5413)
  • [20] -, A remark on Mirimanoff polynomials, Comment. Math. Univ. St. Paul. 31 (1982), 89-97. MR 674586 (84b:10022)
  • [21] -, On the Kummer's system of congruences, Comment. Math. Univ. St. Paul. 35 (1986), 137-163. MR 864734 (87m:11016)
  • [22] -, A note on the index of irregularity, J. Number Theory 22 (1986), 125-138. MR 826946 (87d:11080)
  • [23] -, Fermat's last theorem and the Fermat quotients, Comment. Math. Univ. St. Paul 41 (1992), 35-54. MR 1166223 (93f:11028)
  • [24] J. W. Tanner and S. S. Wagstaff, New congruences for the Bernoulli numbers, Math. Comp. 48 (1987), 341-350. MR 866120 (87m:11017)
  • [25] T. Uehara, Fermat's conjecture and Bernoulli numbers, Rep. Fac. Sci. Engrg. Saga Univ. Math. 6 (1978), 9-14. MR 490247 (80a:12008)
  • [26] S. S Wagstaff, The irregular primes to 125,000, Math. Comp. 32 (1978), 583-591. MR 0491465 (58:10711)
  • [27] L. C. Washington, Introduction to cyclotomic fields, Springer-Verlag, New York, Heidelberg, and Berlin, 1982. MR 718674 (85g:11001)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1196218-5
Keywords: Kummer system of congruences, the first case of Fermat's Last Theorem, Stickelberger ideal, index of irregularity of a prime
Article copyright: © Copyright 1994 American Mathematical Society

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