Bernoulli Numbers definition of Bernoulli Numbers in the Free Online Encyclopedia.

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 x^p + y^p = z^p 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 B_2m (m = 1, 2,. . .) is written instead of B_m to designate Bernoulli numbers; furthermore, it is assumed that

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

REFERENCES

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.

S. B. STECHKIN

Mentioned in	?		References in periodicals archive	?		Encyclopedia browser	?		Full browser	?
No references found	2] + [1/2]t, and the first few Bernoulli numbers are [B. Improvements and generalizations of some Euler Gruss type ... by Hwang, Dah-Yan; Hwang, Shiow-Ru; Wang, Chung-Shin / Tamsui Oxford Journal of Mathematical Sciences Although the Bernoulli numbers formula had previously been solved, Altourmaimi's solution was praised because of his young age and school level, the paper reported. Iraq-born teen solves 300-year old math puzzle by Al Arabiya (Saudi Arabia) In just four months, Mohamed Altoumaimi has found a formula to explain and simplify the so-called Bernoulli numbers, a sequence of calculations named after the 17th century Swiss mathematician Jacob Bernoulli, the Dagens Nyheter daily said. Iraq-born teen cracks maths puzzle by AFP / AFP Global Edition	Bernhardi, Friedrich von Bernhardi, Theodor Von Bernhardt Bernhardt, Sarah Bernhardus Varenius Bernheim Petition Berni, Antonio Berni, Francesco Bernice State Park Bernicia Bernina Bernina Pass Bernini Bernini, Gian Lorenzo Bernini, Giovanni Lorenzo Bernoulli Bernoulli Box Bernoulli differential equation Bernoulli distribution Bernoulli effect Bernoulli Equation Bernoulli experiments Bernoulli family Bernoulli law Bernoulli number Bernoulli Numbers Bernoulli polynomial Bernoulli principle Bernoulli theorem Bernoulli trials Bernoulli's Equation Bernoulli's lemniscate Bernoulli's principle Bernoulli's Scheme Bernoulli's Theorem Bernoulli-Euler law Bernshtam, Aleksandr Bernshtam, Aleksandr Natanovich Bernshtein, Nikolai Bernshtein, Nikolai Aleksandrovich Bernshtein, Sergei Bernshtein, Sergei Ignatevich Bernshtein, Sergei Natanovich Bernstein Bernstein condition Bernstein, Aline Bernstein, Baroness Bernstein, Carl Bernstein, Eduard Bernstein, Elmer Bernstein, Leonard	Bernische Kraftwerke Bernischer Blindenfürsorgeverein Bernischer Kantonal Musikverband Bernoilli equation Bernoilli equation Bernoilli equation Bernouilli Bernouilli Bernouilli number Bernouilli numbers Bernouli random variable Bernouli random variable Bernoulli Bernoulli Bernoulli Box Bernoulli Box Bernoulli differential equation Bernoulli Disk Drive Bernoulli distribution Bernoulli distribution Bernoulli distribution Bernoulli effect Bernoulli effect Bernoulli effect Bernoulli equation Bernoulli experiments Bernoulli family Bernoulli law Bernoulli Modulated Markov Process Bernoulli number Bernoulli Numbers Bernoulli polynomial Bernoulli polynomials Bernoulli principal Bernoulli principal Bernoulli principal Bernoulli Principle Bernoulli Principle Bernoulli Principle Bernoulli random variable Bernoulli random variable Bernoulli Society for Mathematical Statistics and Probability Bernoulli Solutions and Systems Network Bernoulli theorem Bernoulli theorem Bernoulli theorem Bernoulli trial Bernoulli trial Bernoulli trial Bernoulli trials Bernoulli trials Bernoulli trials Bernoulli's differential equation Bernoulli's equation Bernoulli's equation Bernoulli's equation Bernoulli's law Bernoulli's law Bernoulli's law Bernoulli's law Bernoulli's lemniscate

Mentioned in

References in periodicals archive

Encyclopedia browser

Full browser

No references found

2] + [1/2]t, and the first few Bernoulli numbers are [B.

Improvements and generalizations of some Euler Gruss type ... by Hwang, Dah-Yan; Hwang, Shiow-Ru; Wang, Chung-Shin / Tamsui Oxford Journal of Mathematical Sciences

Although the Bernoulli numbers formula had previously been solved, Altourmaimi's solution was praised because of his young age and school level, the paper reported.

Iraq-born teen solves 300-year old math puzzle by Al Arabiya (Saudi Arabia)

In just four months, Mohamed Altoumaimi has found a formula to explain and simplify the so-called Bernoulli numbers, a sequence of calculations named after the 17th century Swiss mathematician Jacob Bernoulli, the Dagens Nyheter daily said.

Iraq-born teen cracks maths puzzle by AFP / AFP Global Edition

Bernhardi, Friedrich von

Bernhardi, Theodor Von