In recognition of her additions to Menabrea's paper, which included a way to calculate Bernoulli numbers using the machine, she has been described as the first computer programmer.

In mathematics, the Bernoulli numbers B < sub > n </ sub > are a sequence of rational numbers with deep connections to number theory.

The values of the first few Bernoulli numbers are

If the convention B < sub > 1 </ sub >=− is used, this sequence is also known as the first Bernoulli numbers ( / in OEIS ); with the convention B < sub > 1 </ sub >=+ is known as the second Bernoulli numbers ( / in OEIS ).

Except for this one difference, the first and second Bernoulli numbers agree.

Since B < sub > n </ sub >= 0 for all odd n > 1, and many formulas only involve even-index Bernoulli numbers, some authors write B < sub > n </ sub > instead of B < sub > 2n </ sub >.

The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jakob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa.

Ada Lovelace's note G on the analytical engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine.

As a result, the Bernoulli numbers have the distinction of being the subject of the first computer program.

Bernoulli numbers feature prominently in the closed form expression of the sum of the m-th powers of the first n positive integers.

The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli's formula:

Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers.

Unfortunately in the literature the definition is given in two variants: Despite the fact that Bernoulli defined B < sub > 1 </ sub > = 1 / 2 ( now known as " second Bernoulli numbers "), some authors set B < sub > 1 </ sub > = − 1 / 2 (" first Bernoulli numbers ").

The related sum occurs in the study of Bernoulli numbers ; the harmonic numbers also appear in the study of Stirling numbers.

Bernoulli also wrote a large number of papers on various mechanical questions, especially on problems connected with vibrating strings, and the solutions given by Brook Taylor and by Jean le Rond d ' Alembert.

It is the probability distribution of a certain number of failures and successes in a series of independent and identically distributed Bernoulli trials.

The Fourier series is named in honour of Joseph Fourier ( 1768 – 1830 ), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d ' Alembert, and Daniel Bernoulli.

Bernoulli trials may also lead to negative binomial distributions ( which count the number of successes in a series of repeated Bernoulli trials until a specified number of failures are seen ), as well as various other distributions.

It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability.

be the formal power series with the property that the coefficient of x < sup > n </ sup > in Q ( x )< sup > n + 1 </ sup > is 1 ( where the B < sub > i </ sub > are Bernoulli numbers ).

In other words, if a planet, M, follows an elliptical orbit around a focus F and has a point P where PM is tangent to the curve and FPM is a right angle so that FP is the perpendicular to the tangent, then the centripetal force at point P is proportional to F * M /( R *( F * P )< sup > 3 </ sup >) where R is the radius of the curvature at M. Johann Bernoulli proved this formula in 1710.

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y ( x ) of Bessel's differential equation:

Then the periodic Bernoulli functions P < sub > n </ sub > are defined as

We define the periodic Bernoulli functions P < sub > n </ sub > by

In this way we get a proof of the Euler – Maclaurin summation formula by mathematical induction, in which the induction step relies on integration by parts and on the identities for periodic Bernoulli functions.

A set of functions dual to the Bernoulli polynomials are given by

The Euler – MacLaurin summation formula can thus be seen to be an outcome of the representation of functions on the unit interval by the direct product of the Bernoulli polynomials and their duals.

Friedrich Wilhelm Bessel ( 22 July 1784 – 17 March 1846 ) was a German mathematician, astronomer, and systematizer of the Bessel functions ( which were discovered by Daniel Bernoulli ).

In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function.

In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions.

Moreover, already in 1893 Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers,

Binomial distributions were treated by James Bernoulli about 1700, and for this reason binomial trials are sometimes called Bernoulli trials.

Because these two definitions can be transformed simply by into the other, some formulae have this alternatingly (- 1 )< sup > n </ sup >- term and others not depending on the context, but it is not possible to decide in favor of one of these definitions to be the correct or appropriate or natural one ( for the abstract Bernoulli numbers ).

for the Bernoulli numbers exist.

However, both simple and high-end algorithms for computing Bernoulli numbers exist.

In some applications it is useful to be able to compute the Bernoulli numbers B < sub > 0 </ sub > through B < sub > p − 3 </ sub > modulo p, where p is a prime ; for example to test whether Vandiver's conjecture holds for p, or even just to determine whether p is an irregular prime.

David Harvey describes an algorithm for computing Bernoulli numbers by computing B < sub > n </ sub > modulo p for

Goldbach is most noted for his correspondence with Leibniz, Euler, and Bernoulli, especially in his 1742 letter to Euler stating his Goldbach's conjecture.

− 1 / 30, … are the Bernoulli numbers, and R is an error term which is normally small for suitable values of p and depends on n, m, p and f. ( The formula is often written with the subscript taking only even values, since the odd Bernoulli numbers are zero except for B < sub > 1 </ sub >.

Here we denote with the Bernoulli number of the second kind ( only because the historical reason of formation of this article ) which differ from the first kind only for the index 1.

Note that the Bernoulli numbers are defined as B < sub > n </ sub > = B < sub > n </ sub >( 0 ), and that these vanish for odd n greater than 1.

At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics.

Two years later, Daniel Bernoulli, who was fed up with the censorship and hostility he faced at St. Petersburg, left for Basel.