Bernoulli Number - Arithmetical Properties of The Bernoulli Numbers

Arithmetical Properties of The Bernoulli Numbers

The Bernoulli numbers can be expressed in terms of the Riemann zeta function as Bn = − nζ(1 − n) for integers n ≥ 0 provided for n = 0 and n = 1 the expression − nζ(1 − n) is understood as the limiting value and the convention B1 = 1/2 is used. This intimately relates them to the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties. For example, the Agoh–Giuga conjecture postulates that p is a prime number if and only if pBp−1 is congruent to −1 modulo p. Divisibility properties of the Bernoulli numbers are related to the ideal class groups of cyclotomic fields by a theorem of Kummer and its strengthening in the Herbrand-Ribet theorem, and to class numbers of real quadratic fields by Ankeny–Artin–Chowla.

Read more about this topic:  Bernoulli Number

Famous quotes containing the words properties and/or numbers:

    The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.
    John Locke (1632–1704)

    The only phenomenon with which writing has always been concomitant is the creation of cities and empires, that is the integration of large numbers of individuals into a political system, and their grading into castes or classes.... It seems to have favored the exploitation of human beings rather than their enlightenment.
    Claude Lévi-Strauss (b. 1908)