×

zbMATH — the first resource for mathematics

Irregular primes and cyclotomic invariants. (English) Zbl 0302.10020

MSC:
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11R18 Cyclotomic extensions
11D41 Higher degree equations; Fermat’s equation
11R23 Iwasawa theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. I. Borevich and I. R. Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. · Zbl 0145.04902
[2] L. E. DICKSON, History of the Theory of Numbers. Vol. II, Carnegie Institution of Wash., Washington, D. C., 1920. · JFM 47.0888.08
[3] A. FRIEDMANN & J. TAMARKINE, ”Quelques formules conçernant la théorie de la fonction [x] et des nombres de Bernoulli,” J. Reine Angew. Math., v. 135, 1909, pp. 146-156.
[4] Kenkichi Iwasawa, On \Gamma -extensions of algebraic number fields, Bull. Amer. Math. Soc. 65 (1959), 183 – 226. · Zbl 0089.02402
[5] Kenkichi Iwasawa, On the \?-invariants of cyclotomic fields, Acta Arith. 21 (1972), 99 – 101. · Zbl 0253.12012
[6] K. IWASAWA, Lecture Notes of a Course at Princeton, Fall semester, 1971.
[7] Kenkichi Iwasawa and Charles C. Sims, Computation of invariants in the theory of cyclotomic fields, J. Math. Soc. Japan 18 (1966), 86 – 96. · Zbl 0141.04901 · doi:10.2969/jmsj/01810086 · doi.org
[8] Wells Johnson, On the vanishing of the Iwasawa invariant \?_\? for \?<8000, Math. Comp. 27 (1973), 387 – 396. · Zbl 0281.12006
[9] Wells Johnson, Irregular prime divisors of the Bernoulli numbers, Math. Comp. 28 (1974), 653 – 657. · Zbl 0293.10008
[10] V. V. Kobelev, A proof of Fermat’s theorem for all prime ewponents less that 5500., Dokl. Akad. Nauk SSSR 190 (1970), 767 – 768 (Russian).
[11] D. H. LEHMER, ”Automation and pure mathematics” in Applications of Digital Computers, W. F. Freiberger and W. Prager, editors, Ginn, Boston, Mass., 1963.
[12] D. H. Lehmer, Emma Lehmer, and H. S. Vandiver, An application of high-speed computing to Fermat’s last theorem, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), 25 – 33. · Zbl 0055.04004
[13] Tauno Metsänkylä, Note on the distribution of irregular primes, Ann. Acad. Sci. Fenn. Ser. A I No. 492 (1971), 7. · Zbl 0208.05502
[14] T. Metsänkylä, Class numbers and \?-invariants of cyclotomic fields, Proc. Amer. Math. Soc. 43 (1974), 299 – 300. · Zbl 0257.12004
[15] Tauno Metsänkylä, Distribution of irregular prime numbers, J. Reine Angew. Math. 282 (1976), 126 – 130. · Zbl 0324.10041 · doi:10.1515/crll.1976.282.126 · doi.org
[16] Hugh L. Montgomery, Distribution of irregular primes, Illinois J. Math. 9 (1965), 553 – 558. · Zbl 0131.04501
[17] J. L. Selfridge, C. A. Nicol, and H. S. Vandiver, Proof of Fermat’s last theorem for all prime exponents less than 4002, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 970 – 973. · Zbl 0065.27304
[18] J. L. SELFRIDGE & B. W. POLLACK, ”Fermat’s last theorem is true for any exponent up to 25,000,” Notices Amer. Math. Soc., v. 11, 1964, p. 97. Abstract #608-138.
[19] Carl Ludwig Siegel, Zu zwei Bemerkungen Kummers, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1964 (1964), 51 – 57 (German). · Zbl 0119.27701
[20] H. S. VANDIVER, ”On Kummer’s memoir of 1857 concerning Fermat’s last theorem,” Proc. Nat. Acad. Sci U.S.A., v. 6, 1920, pp. 266-269. · JFM 47.0151.01
[21] H. S. VANDIVER, ”On the class number of the field \( \Omega ({e^{2i\pi /{p^n}}})\) and the second case of Fermat’s last theorem,” Proc. Nat. Acad. Sci. U.S.A., v. 6, 1920, pp. 416-421.
[22] H. S. Vandiver, On Fermat’s last theorem, Trans. Amer. Math. Soc. 31 (1929), no. 4, 613 – 642. · JFM 55.0701.04
[23] H. S. Vandiver, On Bernoulli’s numbers and Fermat’s last theorem, Duke Math. J. 3 (1937), no. 4, 569 – 584. · Zbl 0018.00505 · doi:10.1215/S0012-7094-37-00345-4 · doi.org
[24] H. S. Vandiver, Examination of methods of attack on the second case of Fermat’s last theorem, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), 732 – 735. · Zbl 0056.04103
[25] H. S. Vandiver, On developments in an arithmetic theory of the Bernoulli and allied numbers, Scripta Math. 25 (1961), 273 – 303. · Zbl 0100.26901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.