New congruences for the Bernoulli numbers
Authors:
Jonathan W. Tanner and Samuel S. Wagstaff
Journal:
Math. Comp. 48 (1987), 341350
MSC:
Primary 11B68; Secondary 11D41, 11Y50
MathSciNet review:
866120
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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.
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 R. W. Hockney & C. R. Jesshope, Parallel Computers: Architecture, Programming and Algorithms, Adam Hilger, Bristol, 1981.
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 Wells Johnson, "padic proofs of congruences for the Bernoulli numbers," J. Number Theory, v. 7, 1975, pp. 251265. MR 0376512 (51:12687)
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 D. H. Lehmer, "Lacunary recurrence formulas for the numbers of Bernoulli and Euler," Ann. of Math., v. 36, 1935, pp. 637649. MR 1503241
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 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. 569584. MR 1546011
 [6]
 Samuel S. Wagstaff, Jr., "The irregular primes to 125000," Math. Comp., v. 32, 1978, pp. 583591. MR 0491465 (58:10711)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198708661204
PII:
S 00255718(1987)08661204
Keywords:
Bernoulli numbers,
Vandiver's congruence,
Fermat's "Last Theorem"
Article copyright:
© Copyright 1987
American Mathematical Society
