Bernoulli number

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In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers.

Bn = 0 for all odd n other than 1. B1 = 1/2 or −1/2 depending on the convention adopted. The values of the first few nonzero Bernoulli numbers are (more values below):

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. Seki's discovery was posthumously published in 1712[1][2] in his work Katsuyo Sampo; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713.

They appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

In note G of Ada Lovelace's notes on the analytical engine from 1842, Lovelace describes an algorithm for generating Bernoulli numbers with Babbage's machine.[3] As a result, the Bernoulli numbers have the distinction of being the subject of the first computer program.


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