Irregular primes and cyclotomic invariants

Author:
Wells Johnson

Journal:
Math. Comp. **29** (1975), 113-120

MSC:
Primary 12A35; Secondary 10A40, 10B15

DOI:
https://doi.org/10.1090/S0025-5718-1975-0376606-9

MathSciNet review:
0376606

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Abstract | References | Similar Articles | Additional Information

Abstract: The table of irregular primes less than 30000 has been computed and deposited in the UMT file. The fraction of irregular primes in this range is 0.3924, close to the heuristic prediction of . Fermat's Last Theorem has been verified for all prime exponents , and the cyclotomic invariants , and of Iwasawa have been completely determined for these primes. The computations show that for *p* in this range, and the invariants and both equal the index of irregularity of *p*.

**[1]**Z. I. BOREVIČ & I. R. ŠAFAREVIČ,*Number Theory*, "Nauka", Moscow, 1964; English transl., Pure and Appl. Math., vol. 20, Academic Press, New York, 1966. MR**30**#1080;**33**#4001. MR**0195803 (33:4001)****[2]**L. E. DICKSON,*History of the Theory of Numbers*. Vol. II, Carnegie Institution of Wash., Washington, D. C., 1920.**[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]**K. IWASAWA, "On -extensions of algebraic number fields,"*Bull. Amer. Math. Soc.*, v. 65, 1959, pp. 183-226. MR**23**#A1630. MR**0124316 (23:A1630)****[5]**K. IWASAWA, "On the -invariants of cyclotomic fields,"*Acta Arith.*, v. 21, 1972, pp. 99-101. MR**46**#1750. MR**0302606 (46:1750)****[6]**K. IWASAWA,*Lecture Notes of a Course at Princeton*, Fall semester, 1971.**[7]**K. IWASAWA &. C. SIMS, "Computation of invariants in the theory of cyclotomic fields,"*J. Math. Soc. Japan*, v. 18, 1966, pp. 86-96. MR**34**#2560. MR**0202700 (34:2560)****[8]**W. JOHNSON, "On the vanishing of the Iwasawa invariant for ,"*Math. Comp.*, v. 27, 1973, pp. 387-396. MR**0384748 (52:5621)****[9]**W. JOHNSON, "Irregular prime divisors of the Bernoulli numbers,"*Math. Comp.*, v. 28, 1974, pp. 653-657. MR**0347727 (50:229)****[10]**V. V. KOBELEV, "Proof of Fermat's last theorem for all prime exponents less than 5500,"*Dokl. Akad. Nauk SSSR*, v. 190, 1970, pp. 767-768 =*Soviet Math. Dokl.*, v. 11, 1970, pp. 188-190. MR**41**#3363. MR**0258717 (41:3363)****[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, E. LEHMER & H. S. VANDIVER, "An application of high-speed computing to Fermat's last theorem,"*Proc. Nat. Acad. Sci. U.S.A.*, v. 40, 1954, pp. 25-33. MR**15**, 778. MR**0061128 (15:778f)****[13]**T. METSÄNKYLÄ, "Note on the distribution of irregular primes,"*Ann. Acad. Sci. Fenn. Ser. A I*, No. 492, 1971, 7pp. MR**43**#168. MR**0274403 (43:168)****[14]**T. METSÄNKYLÄ, "Class numbers and -invariants of cyclotomic fields,"*Proc. Amer. Math. Soc.*, v. 43, 1974, pp. 299-300. MR**0332721 (48:11047)****[15]**T. METSÄNKYLÄ, "Distribution of irregular prime numbers,"*J. Reine Angew. Math.*(To appear.) MR**0399014 (53:2865)****[16]**H. L. MONTGOMERY, "Distribution of irregular primes,"*Illinois J. Math.*, v. 9, 1965, pp. 553-558. MR**31**#5861. MR**0181633 (31:5861)****[17]**J. L. SELFRIDGE, C. A. NICOL & H. S. VANDIVER, "Proof of Fermat's last theorem for all prime exponents less than 4002,"*Proc. Nat. Acad. Sci. U.S.A.*, v. 41, 1955, pp. 970-973. MR**17**, 348. MR**0072892 (17:348a)****[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]**C. L. SIEGEL, "Zu zwei Bemerkungen Kummers,"*Nachr. Akad. Wiss. Göttingen*, 1964, Nr. 6, 51-57. MR**29**#1198; Also in*Gesammelte Abhandlungen*. Vol. III, Springer-Verlag, New York, 1966, pp. 436-442. MR**0163899 (29:1198)****[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.**[21]**H. S. VANDIVER, "On the class number of the field 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.*, v. 31, 1929, pp. 613-642. MR**1501503****[23]**H. S. VANDIVER, "On Bernoulli's numbers and Fermat's last theorem,"*Duke Math. J.*, v. 3, 1937, pp. 569-584. MR**1546011****[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.*, v. 40, 1954, pp. 732-735. MR**16**, 13. MR**0062758 (16:13f)****[25]**H. S. VANDIVER, "Is there an infinity of regular primes?,"*Scripta Math.*, v. 21, 1955, pp. 306-309. MR**0142497 (26:66)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1975-0376606-9

Keywords:
Irregular primes,
Bernoulli numbers,
Fermat's Last Theorem,
cyclotomic fields,
class numbers,
-extensions,
cyclotomic invariants

Article copyright:
© Copyright 1975
American Mathematical Society