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In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...
Gregory coefficients G n, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, [1] [2] [3] ...
The following removed paragraphs are personal opinions of Peter Luschny, see an essay of Luschny The Bernoulli Confusion. This guy (is he an expert on number theory or a graduate mathematician at all? I don't think so!) rigidly claims, against the consensus of experts, to change the definition of Bernoulli numbers so that B(1) = 1/2.
In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by Karl von Staudt and Thomas Clausen . Specifically, if n is a positive integer and we add 1/ p to the Bernoulli number B 2 n for every prime p such that p − 1 divides 2 n , then we obtain an integer; that is,
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Bernoulli numbers can be calculated in many ways, but Lovelace deliberately chose an elaborate method in order to demonstrate the power of the engine. In Note G, she states: "We will terminate these Notes by following up in detail the steps through which the engine could compute the Numbers of Bernoulli, this being (in the form in which we ...
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The simplest form of Kummer's congruence states that ()where p is a prime, h and k are positive even integers not divisible by p−1 and the numbers B h are Bernoulli numbers.