Irregular prime divisors of the Bernoulli numbers

Author:
Wells Johnson

Journal:
Math. Comp. **28** (1974), 653-657

MSC:
Primary 10A40; Secondary 12A35, 12A50

DOI:
https://doi.org/10.1090/S0025-5718-1974-0347727-0

MathSciNet review:
0347727

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If *p* is an irregular prime, $p < 8000$, then the indices 2*n* for which the Bernoulli quotients ${B_{2n}}/2n$ are divisible by ${p^2}$ are completely characterized. In particular, it is always true that $2n > p$ and that ${B_{2n}}/2n\;\nequiv ({B_{2n + p - 1}}/2n + p - 1)\pmod {p^2}$ if *(p,2n)* is an irregular pair. As a result, we obtain another verification that the cyclotomic invariants ${\mu _p}$ of Iwasawa all vanish for primes $p < 8000$.

- A. I. Borevich and I. R. Shafarevich,
*Number theory*, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR**0195803** - L. Carlitz,
*Note on irregular primes*, Proc. Amer. Math. Soc.**5**(1954), 329â€“331. MR**61124**, DOI https://doi.org/10.1090/S0002-9939-1954-0061124-6 - Kenkichi Iwasawa,
*On some invariants of cyclotomic fields*, Amer. J. Math. 80 (1958), 773-783; erratum**81**(1958), 280. MR**0124317** - Kenkichi Iwasawa and Charles C. Sims,
*Computation of invariants in the theory of cyclotomic fields*, J. Math. Soc. Japan**18**(1966), 86â€“96. MR**202700**, DOI https://doi.org/10.2969/jmsj/01810086 - Wells Johnson,
*On the vanishing of the Iwasawa invariant $\mu _{p}$ for $p<8000$*, Math. Comp.**27**(1973), 387â€“396. MR**384748**, DOI https://doi.org/10.1090/S0025-5718-1973-0384748-5 - Emma Lehmer,
*On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson*, Ann. of Math. (2)**39**(1938), no. 2, 350â€“360. MR**1503412**, DOI https://doi.org/10.2307/1968791 - Tauno MetsĂ¤nkylĂ¤,
*Note on the distribution of irregular primes*, Ann. Acad. Sci. Fenn. Ser. A I No.**492**(1971), 7. MR**0274403** - Hugh L. Montgomery,
*Distribution of irregular primes*, Illinois J. Math.**9**(1965), 553â€“558. MR**181633** - F. Pollaczek,
*Ăśber die irregulĂ¤ren KreiskĂ¶rper der $l$-ten und $l^2$-ten Einheitswurzeln*, Math. Z.**21**(1924), no. 1, 1â€“38 (German). MR**1544682**, DOI https://doi.org/10.1007/BF01187449
J. Uspensky & M. Heaslet,

*Elementary Number Theory*, McGraw-Hill, New York, 1939. MR

**1**, 38. H. S. Vandiver, "Is there an infinity of regular primes?,"

*Scripta Math.*, v. 21, 1955, pp. 306-309.

Retrieve articles in *Mathematics of Computation*
with MSC:
10A40,
12A35,
12A50

Retrieve articles in all journals with MSC: 10A40, 12A35, 12A50

Additional Information

Keywords:
Bernoulli numbers,
irregular primes,
cyclotomic invariants

Article copyright:
© Copyright 1974
American Mathematical Society