Bernoulli number

The Bernoulli numbers come from the coefficients in the Taylor expansion of x/(e^x-1). They can be defined recursively by setting B₀=1, and then using

The first few Bernoulli numbers are B₀=1, B₁=-1/2, B₂=1/6, B₃=0, B₄=-1/30, B₅=0, B₆=1/42, B₇=0, B₈=-1/30, B₉=0, and B₁₀=5/66. Notice that all of the odd terms, B_2n+1 (n > 1), are zero; and the even terms alternate in sign.

These numbers can also be defined using the Riemann zeta function as follows

Finally, using Stirling's formula, we have

The Bernoulli numbers first appeared in the posthumous work "Ars Conjectandi" (1713) by Jakob Bernoulli. Euler used them to express the sums of equal powers of consecutive integers. They also are important in classical assaults of Fermat's Last Theorem.

See Also: Regular

Related pages (outside of this work)

• A Bibliography of Bernoulli Numbers (excellent!)

References:

Ribenboim95 (pp. 217-218)

P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995.  pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]