There are two definitions for the Bernoulli numbers. The older one, no longer in widespread use, defines the
Bernoulli numbers by the equations

(1) |

for , or

(2) | |||

(3) |

for (Whittaker and Watson 1990, p. 125). Gradshteyn and Ryzhik (1979) denote these numbers , while Bernoulli numbers defined by the newer (National Bureau of Standards) definition are denoted . The Bernoulli numbers may be calculated from the integral

(4) |

(5) |

The first few Bernoulli numbers are

Bernoulli numbers defined by the modern definition are denoted and also called ``Even-index'' Bernoulli numbers. These are
the Bernoulli numbers returned by the *Mathematica*
(Wolfram Research, Champaign, IL) function `BernoulliB[n]`. These Bernoulli numbers are a superset of the archaic ones since

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

or

(12) |

(13) |

(14) |

(15) |

(16) |

The Bernoulli numbers satisfy the identity

(17) |

(18) |

(19) |

(20) |

The Denominator of is given by the von Staudt-Clausen Theorem

(21) |

(Sloane's A000367 and A002445). In addition,

(22) |

Bernoulli first used the Bernoulli numbers while computing
. He used the property of the Figurate
Number Triangle that

(23) |

(24) |

(25) |

G. J. Fee and S. Plouffe have computed , which has Digits (Plouffe). Plouffe and collaborators have also calculated for up to 72,000.

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula.''
§23.1 in *Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables, 9th printing.* New York: Dover, pp. 804-806, 1972.

Arfken, G. ``Bernoulli Numbers, Euler-Maclaurin Formula.'' §5.9 in
*Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 327-338,
1985.

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations and Essays, 13th ed.*
New York: Dover, p. 71, 1987.

Berndt, B. C. *Ramanujan's Notebooks, Part IV.* New York: Springer-Verlag, pp. 81-85, 1994.

Boyer, C. B. *A History of Mathematics.* New York: Wiley, 1968.

Castellanos, D. ``The Ubiquitous Pi. Part I.'' *Math. Mag.* **61**, 67-98, 1988.

Conway, J. H. and Guy, R. K. In *The Book of Numbers.* New York: Springer-Verlag, pp. 107-110, 1996.

Gradshteyn, I. S. and Ryzhik, I. M. *Tables of Integrals, Series, and Products, 5th ed.*
San Diego, CA: Academic Press, 1980.

Hardy, G. H. and Wright, W. M. *An Introduction to the Theory of Numbers, 5th ed.* Oxford, England: Oxford
University Press, pp. 91-93, 1979.

Ireland, K. and Rosen, M. ``Bernoulli Numbers.'' Ch. 15 in *A Classical Introduction to Modern Number
Theory, 2nd ed.* New York: Springer-Verlag, pp. 228-248, 1990.

Knuth, D. E. and Buckholtz, T. J. ``Computation of Tangent, Euler, and Bernoulli Numbers.'' *Math. Comput.*
**21**, 663-688, 1967.

Plouffe, S. ``Plouffe's Inverter: Table of Current Records for the Computation of Constants.'' http://www.lacim.uqam.ca/pi/records.html.

Ramanujan, S. ``Some Properties of Bernoulli's Numbers.'' *J. Indian Math. Soc.* **3**, 219-234, 1911.

Sloane, N. J. A. Sequences
A000367/M4039
and A002445/M4189
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Spanier, J. and Oldham, K. B. ``The Bernoulli Numbers, .''
Ch. 4 in *An Atlas of Functions.* Washington, DC: Hemisphere, pp. 35-38, 1987.

Wagstaff, S. S. Jr. ``Ramanujan's Paper on Bernoulli Numbers.'' *J. Indian Math. Soc.* **45**, 49-65, 1981.

Whittaker, E. T. and Watson, G. N. *A Course in Modern Analysis, 4th ed.* Cambridge, England:
Cambridge University Press, 1990.

© 1996-9

1999-05-26