Bernoulli Numbers

Bernoulli Numbers


a special sequence of rational numbers which figures in various problems of mathematical analysis and the theory of numbers. The values of the first six Bernoulli numbers are

In mathematical analysis, Bernoulli numbers appear as the coefficients of expansion of certain elementary functions in power series—for example,

The Euler-Maclaurin summation formula is one of the most important formulas in which Bernoulli numbers are encountered. The sums of many series and the values of improper integrals are expressed in terms of Bernoulli numbers. Bernoulli numbers first appeared in the posthumous work of Jakob Bernoulli (1713) in connection with the calculation of the sum of identical powers of natural numbers. He proved that

Recurrence formulas that permit the sequential calculation of Bernoulli numbers, as well as explicit formulas (which have a rather complex form), are known for Bernoulli numbers.

There is great interest in the theoretic-numerical properties of the Bernoulli numbers. In 1850 the German mathematician E. Kummer established that Fermat’s equation xp + yp = zp is not solved in integers x, y, and z which are not zero unless a prime number p > 2 divides the numerators of the Bernoulli numbers B1, B2, . . ., B(p - 3)/2. Often (-1)m-1 B2m (m = 1, 2,. . .) is written instead of Bm to designate Bernoulli numbers; furthermore, it is assumed that

B0 = 1, B1 = -½, B3 = B5 = B7 = . . . = 0


Chistiakov, I. I. Bernullievye chisla. Moscow, 1895.
Kudriavtsev, V. A. Summirovanie stepenei chisel natural’nogo riada i chisla Bernulli. Moscow-Leningrad, 1936.
Whittaker, E. T., and G. N. Watson. Kurs sovremennogo analiza, 2nd ed., part 1. Moscow, 1963. (Translated from English.)
Landau, E. Vorlesungenüber Zahlentheorie, vol. 3. New York, 1927.


References in periodicals archive ?
For the second edition he has corrected errors; clarified some confusion; and added new problems on Bernoulli numbers, metric spaces, and differential equations.
represent the Bernoulli numbers 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, .
A note on the denominators of Takao KOMATSU, Florian LUCA Bernoulli numbers and Claudio DE J.
They included a way to calculate Bernoulli numbers (a mathematical sequence of numbers) using the machine.
Determinant expressions for q-harmonic congruences and degenerate Bernoulli numbers.
k](0), k[greater than or equal to]0, the Bernoulli numbers.
In just four months, Mohamed Altoumaimi found a formula to explain and simplify the so-called Bernoulli numbers, a sequence of calculations connected to numbe theory named after the 17th century Swiss mathematician Jacob Bernoulli, the Dagens Nyheter daily said.
Faulhaber was an excellent mathematician, who among other things published the first 16 Bernoulli numbers in 1631 [18, p.
Stein appeals to graduates, advanced undergraduates and non-specialists in number theory as be describes the modular forms of weights and levels, Dirichlet characters, Eisenstein series and Bernoulli numbers, dimensions formulas, linear algebra, general modular symbols, computing with newforms, and computing periods.
He'd written a paper about Bernoulli numbers and I realised that I had noticed some patterns which even he hadn't spotted.
The historical Lady Byron, sometimes called the world's first programmer, developed a program for Babbage's machine to calculate the complex sequence called Bernoulli numbers.
3) yields the identity showed by Euler for sums of products of Bernoulli numbers