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

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

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*, v. 7, 1975, pp. 251-265. MR**0376512 (51:12687)****[3]**D. H. Lehmer, "Lacunary recurrence formulas for the numbers of Bernoulli and Euler,"*Ann. of Math.*, v. 36, 1935, pp. 637-649. MR**1503241****[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.*, v. 3, 1937, pp. 569-584. MR**1546011****[6]**Samuel S. Wagstaff, Jr., "The irregular primes to 125000,"*Math. Comp.*, v. 32, 1978, pp. 583-591. MR**0491465 (58:10711)**

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Additional Information

DOI:
https://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