# Bernoulli numbers

The sequence of rational numbers discovered by Jacob Bernoulli [1] in connection with the calculation of the sum of equal powers of natural numbers:

The values of the first Bernoulli numbers are:

All odd-indexed Bernoulli numbers except for are zero, and the signs of alternate. Bernoulli numbers are the values of the Bernoulli polynomials at : ; they also often serve as the coefficients of the expansions of certain elementary functions into power series. Thus, for example,

(the so-called generating function of the Bernoulli numbers);

L. Euler in 1740 pointed out the connection between Bernoulli numbers and the values of the Riemann zeta-function for even :

Bernoulli numbers are used to express many improper integrals, such as

Certain relationships involving Bernoulli numbers are:

(the recurrence formula);

The estimates:

hold. Extensive tables of Bernoulli numbers are available; for instance, [2] contains accurate values of for and approximate values for .

Bernoulli numbers have found many applications in mathematical analysis, number theory and approximate calculations.

#### References

[1] | J. Bernoulli, "Ars conjectandi" , Werke , 3 , Birkhäuser (1975) pp. 107–286 (Original: Basle, 1713) |

[2] | H.T. Davis, "Tables of the higher mathematical functions" , 2 , Bloomington (1935) |

[3] | L. Saalschuetz, "Vorlesungen über die Bernoullischen Zahlen" , Berlin (1893) |

[4] | I.I. Chistyakov, "Bernoulli numbers" , Moscow (1895) (In Russian) |

[5] | N. Nielsen, "Traité élémentaire de nombres de Bernoulli" , Paris (1923) |

[6] | V.A. Kudryavtsev, "Summation of powers of natural numbers and Bernoulli numbers" , Moscow-Leningrad (1936) (In Russian) |

[7] | N.E. Nörlund, "Volesungen über Differenzenrechnung" , Springer (1924) |

[8] | A.O. [A.O. Gel'fond] Gelfond, "Differenzenrechnung" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) |

[9] | G.H. Hardy, "Divergent series" , Clarendon Press (1949) |

#### Comments

Bernoulli numbers play an important role in the theory of cyclotomic fields and Fermat's last theorem, see [a1], pp. 40-41, and [a2]. E.g., if is an odd prime number that does not divide the numerators of , then has no solutions in . (See also Cyclotomic field; Fermat great theorem.)

#### References

[a1] | S. Lang, "Cyclotomic fields" , Springer (1978) |

[a2] | P. Ribenboim, "Thirteen lectures on Fermat's last theorem" , Springer (1979) |

**How to Cite This Entry:**

Bernoulli numbers. Yu.N. Subbotin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Bernoulli_numbers&oldid=15872