New congruences for the Bernoulli numbers

Authors:
Jonathan W. Tanner and Samuel S. Wagstaff

Journal:
Math. Comp. **48** (1987), 341-350

MSC:
Primary 11B68; Secondary 11D41, 11Y50

MathSciNet review:
866120

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove a new congruence for computing Bernoulli numbers modulo a prime. Since it is similar to Vandiver's congruences but has fewer terms, it may be used to test primes for regularity efficiently. We have programmed this test on a CYBER 205 computer. Fermat's "Last Theorem" has been proved for all exponents up to 150000.

**[1]**R. W. Hockney & C. R. Jesshope,*Parallel Computers*:*Architecture, Programming and Algorithms*, Adam Hilger, Bristol, 1981.**[2]**Wells Johnson,*𝑝-adic proofs of congruences for the Bernoulli numbers*, J. Number Theory**7**(1975), 251–265. MR**0376512****[3]**D. H. Lehmer,*Lacunary recurrence formulas for the numbers of Bernoulli and Euler*, Ann. of Math. (2)**36**(1935), no. 3, 637–649. MR**1503241**, 10.2307/1968647**[4]**Jonathan W. Tanner,*Proving Fermat's Last Theorem for Many Exponents by Computer*, B. A. Thesis, Harvard University, 1985.**[5]**H. S. Vandiver,*On Bernoulli’s numbers and Fermat’s last theorem*, Duke Math. J.**3**(1937), no. 4, 569–584. MR**1546011**, 10.1215/S0012-7094-37-00345-4**[6]**Samuel S. Wagstaff Jr.,*The irregular primes to 125000*, Math. Comp.**32**(1978), no. 142, 583–591. MR**0491465**, 10.1090/S0025-5718-1978-0491465-4

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1987-0866120-4

Keywords:
Bernoulli numbers,
Vandiver's congruence,
Fermat's "Last Theorem"

Article copyright:
© Copyright 1987
American Mathematical Society