The irregular primes to
Author:
Samuel S. Wagstaff
Journal:
Math. Comp. 32 (1978), 583591
MSC:
Primary 10A40; Secondary 10B15, 12A35
MathSciNet review:
0491465
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We have determined the irregular primes below 125000 and tabulated their distribution. Two primes of index five of irregularity were found, namely 78233 and 94693. Fermat's Last Theorem has been verified for all exponents up to 125000. We computed the cyclotomic invariants , , , and found that for all . The complete factorizations of the numerators of the Bernoulli numbers for and of the Euler numbers for are given.
 [1]
N.
C. Ankeny and S.
Chowla, A further note on the class number of real quadratic
fields, Acta Arith. 7 (1961/1962), 271–272. MR 0137697
(25 #1147)
 [2]
J. BERTRAND, Personal communication.
 [3]
H. T. DAVIS, Tables of the Mathematical Functions, v. II, The Principia Press, San Antonio, 1935.
 [4]
R.
Ernvall and T.
Metsänkylä, Cyclotomic invariants and
𝐸irregular primes, Math. Comp.
32 (1978), no. 142, 617–629. MR 482273
(80c:12004a), http://dx.doi.org/10.1090/S00255718197804822739
 [5]
Kenkichi
Iwasawa, On Γextensions of algebraic
number fields, Bull. Amer. Math. Soc. 65 (1959), 183–226.
MR
0124316 (23 #A1630), http://dx.doi.org/10.1090/S000299041959103177
 [6]
Kenkichi
Iwasawa and Charles
C. Sims, Computation of invariants in the theory of cyclotomic
fields, J. Math. Soc. Japan 18 (1966), 86–96.
MR
0202700 (34 #2560)
 [7]
John
Johnsen, On the distribution of irregular primes, J. Number
Theory 8 (1976), no. 4, 434–437. MR 0432564
(55 #5552)
 [8]
Wells
Johnson, On the vanishing of the Iwasawa
invariant 𝜇_{𝑝} for 𝑝<8000, Math. Comp. 27 (1973), 387–396. MR 0384748
(52 #5621), http://dx.doi.org/10.1090/S00255718197303847485
 [9]
Wells
Johnson, Irregular prime divisors of the
Bernoulli numbers, Math. Comp. 28 (1974), 653–657. MR 0347727
(50 #229), http://dx.doi.org/10.1090/S00255718197403477270
 [10]
Wells
Johnson, Irregular primes and cyclotomic
invariants, Math. Comp. 29 (1975), 113–120.
Collection of articles dedicated to Derrick Henry Lehmer on the occasion of
his seventieth birthday. MR 0376606
(51 #12781), http://dx.doi.org/10.1090/S00255718197503766069
 [11]
D.
H. Lehmer, Emma
Lehmer, and H.
S. Vandiver, An application of highspeed computing to
Fermat’s last theorem, Proc. Nat. Acad. Sci. U. S. A.
40 (1954), 25–33. MR 0061128
(15,778f)
 [12]
Emma
Lehmer, On congruences involving Bernoulli numbers and the
quotients of Fermat and Wilson, Ann. of Math. (2) 39
(1938), no. 2, 350–360. MR
1503412, http://dx.doi.org/10.2307/1968791
 [13]
Tauno
Metsänkylä, Distribution of irregular prime numbers,
J. Reine Angew. Math. 282 (1976), 126–130. MR 0399014
(53 #2865)
 [14]
Michael
A. Morrison and John
Brillhart, A method of factoring and the
factorization of 𝐹₇, Math.
Comp. 29 (1975),
183–205. Collection of articles dedicated to Derrick Henry Lehmer on
the occasion of his seventieth birthday. MR 0371800
(51 #8017), http://dx.doi.org/10.1090/S00255718197503718005
 [15]
M. OHM, "Etwas über die Bernoullischen Zahlen," J. Reine Angew. Math., v. 20, 1840, pp. 1112.
 [16]
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 #608138.
 [17]
J. L. SELFRIDGE & M. WUNDERLICH, Personal communication.
 [18]
Carl
Ludwig Siegel, Zu zwei Bemerkungen Kummers, Nachr. Akad. Wiss.
Göttingen Math.Phys. Kl. II 1964 (1964), 51–57
(German). MR
0163899 (29 #1198)
 [19]
H.
S. Vandiver, On Bernoulli’s numbers and Fermat’s last
theorem, Duke Math. J. 3 (1937), no. 4,
569–584. MR
1546011, http://dx.doi.org/10.1215/S0012709437003454
 [20]
H. WADA, Personal communication.
 [21]
K. WOOLDRIDGE, Some Results in Arithmetical Functions Similar to Euler's PhiFunction, Ph.D. thesis, University of Illinois at UrbanaChampaign, 1975.
 [22]
Itaru
Yamaguchi, On a Bernoulli numbers conjecture, J. Reine Angew.
Math. 288 (1976), 168–175. MR 0424669
(54 #12628)
 [23]
Hideo
Yokoi, On the distribution of irregular primes, J. Number
Theory 7 (1975), 71–76. MR 0364130
(51 #385)
 [1]
 N. C. ANKENY & S. CHOWLA, "A further note on the class number of real quadratic fields," Acta Arith., v. 7, 1962, pp. 271272. MR 0137697 (25:1147)
 [2]
 J. BERTRAND, Personal communication.
 [3]
 H. T. DAVIS, Tables of the Mathematical Functions, v. II, The Principia Press, San Antonio, 1935.
 [4]
 R. ERNVALL & T. METSÄNKYLÄ, "Cyclotomic invariants and Eirregular primes," Math. Comp., v. 32, 1978, pp. later. MR 482273 (80c:12004a)
 [5]
 K. IWASAWA, "On extensions of algebraic number fields," Bull. Amer. Math. Soc., v. 65, 1959, pp. 183226. MR 23 #A1630. MR 0124316 (23:A1630)
 [6]
 K. IWASAWA & C. C. SIMS, "Computation of invariants in the theory of cyclotomic fields," J. Math. Soc. Japan, v. 18, 1966, pp. 8696. MR 34 #2560. MR 0202700 (34:2560)
 [7]
 J. JOHNSEN, "On the distribution of irregular primes," J. Number Theory, v. 8, 1976, pp. 434437. MR 0432564 (55:5552)
 [8]
 W. JOHNSON, "On the vanishing of the Iwasawa invariant for ," Math. Comp., v. 27, 1973, pp. 387396. MR 52 #5621. MR 0384748 (52:5621)
 [9]
 W. JOHNSON, "Irregular prime divisors of the Bernoulli numbers," Math. Comp., v. 28, 1974, pp. 653657. MR 50 #229. MR 0347727 (50:229)
 [10]
 W. JOHNSON, "Irregular primes and cyclotomic invariants," Math. Comp., v. 29, 1975, pp. 113120. MR 51 #12781. MR 0376606 (51:12781)
 [11]
 D. H. LEHMER, E. LEHMER & H. S. VANDIVER, "An application of highspeed computing to Fermat's last theorem," Proc. Nat. Acad. Sci. U.S.A., v. 40, 1954, pp. 2533. MR 15, 778. MR 0061128 (15:778f)
 [12]
 E. LEHMER, "On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson," Ann. of Math., v. 39, 1938, pp. 350360. MR 1503412
 [13]
 T. METSANKYLA, "Distribution of irregular prime numbers," J. Reine Angew. Math., v. 282, 1976, pp. 126130. MR 0399014 (53:2865)
 [14]
 M. A. MORRISON & J. BRILLHART, "A method of factoring and the factorization of ," Math. Comp., v. 29, 1975, pp. 183205. MR 0371800 (51:8017)
 [15]
 M. OHM, "Etwas über die Bernoullischen Zahlen," J. Reine Angew. Math., v. 20, 1840, pp. 1112.
 [16]
 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 #608138.
 [17]
 J. L. SELFRIDGE & M. WUNDERLICH, Personal communication.
 [18]
 C. L. SIEGEL, "Zu zwei Bemerkungen Kummers," Nachr. Akad. Wiss. Göttingen Math.Phys. Kl. II, Nr. 6, 1964, pp. 5157. MR 29 #1198; Also in Gesammelte Abhandlungen, v. III, SpringerVerlag, Berlin and New York, 1966, pp. 436442. MR 0163899 (29:1198)
 [19]
 H. S. VANDIVER, "On Bernoulli's numbers and Fermat's last theorem," Duke Math. J., v. 3, 1937, pp. 569584. MR 1546011
 [20]
 H. WADA, Personal communication.
 [21]
 K. WOOLDRIDGE, Some Results in Arithmetical Functions Similar to Euler's PhiFunction, Ph.D. thesis, University of Illinois at UrbanaChampaign, 1975.
 [22]
 I. YAMAGUCHI, "On a Bernoulli numbers conjecture," J. Reine Angew. Math., v. 288, 1976, pp. 168175. MR 0424669 (54:12628)
 [23]
 H. YOKOI, "On the distribution of irregular primes," J. Number Theory, v. 7, 1975, pp. 7176. MR 0364130 (51:385)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
10A40,
10B15,
12A35
Retrieve articles in all journals
with MSC:
10A40,
10B15,
12A35
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197804914654
PII:
S 00255718(1978)04914654
Keywords:
Bernoulli numbers,
Euler numbers,
irregular primes,
Fermat's Last Theorem,
cyclotomic invariants
Article copyright:
© Copyright 1978
American Mathematical Society
