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.


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The first five Bernoulli numbers with even indices are
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Also, he proved that the multiple zeta function interpolates multiple Bernoulli numbers at negative integers.