Bernoulli Number)

 Set b0 = 1. When k> 0, define

These Bi (I = 0, 1 ,..., K) is called the bernuoli number. By definition, we can naturally see b1 =-, b2 =, B3 = 0, B4 =-, B5 = 0, B6 =, B7 = 0, B8 = -,.... Bernuoli is the number introduced by Jacob bernuoli, a Swiss mathematician, from his book "guessing" (1713 ). In addition to B1, when K is an odd number, BK = 0; when K is an even number, B2, B6, B10 ,... Is a positive score; B4, B8, B12 ,... Is a negative score. Jacob bernuoli introduced bernuoli's number to solve the so-called "Equality power sum" problem:

SK (n) = 1 K + 2 K +... + NK

For S1 (n) = 1 + 2 + 3 +... + N = n (n + 1)

S2 (n) = 12 + 22 + 32 +... + N2 = n (n + 1) (2n + 1 ),

S3 (n) = 13 + 23 + 33 +... + N3 = [n (n + 1)] 2 = N4 + N3 + N2,

S4 (n) = L4 + 24 + 34 +... + N4 = n (n + 1) (2n + 1) (3n2 + 3n-1) = N5 + N4 + n3-n.

To 17th century, has been found to s17 (N), ferma and others from this we can see SK available Sk-1, Sk-2 ,... The Algebraic representation. Generally, when K is an odd number

SK (n) = n (n + 1) × (polynomial of N ),

When K is an even number,

SK (n) = n (n + 1) (2n + 1) × (polynomial of N ).

Finally, it can be proved that SK (n) is a k + 1 polynomial of N.

SK (n) = A1N + a2n2 +... Ak + 1nk + 1

But how can we find these coefficients A1, A2 ,..., What about aK + 1? Jacob bernuoli has obtained the regularity between coefficients and obtained the specific expression of coefficients. The key sequence BK is called the bernuoli number. He gives a formal formula.

SK (n) =,

Note that this is B k + 1 ≡ B k + 1, not a power, but a form notation. According to this

(K + 1) SK (n) = n k + 1 + () b1n K + () B2nk-1 +... + () Bkn.

 Once the bernuoli number is determined, the idempotence problem is solved, and the bernuoli number can be promoted, as defined

The bn in is the bernuoli number, where | x | <2 π. Bernuoli numbers are useful in number theory. For example

X2-py2 =-4 (p limit 1 (mod 4) is a prime number ),

B147N. C. ankni and aiting had guessed that its solution x0 + y0 met py0. In 1960, model proved that during P limit 5 (mod 8), the above conjecture is equivalent to that the numerator of the bernuoli number is not divisible by P. Later, S. JORA proved the same conclusion for P limit 5 (mod 8. In the proof of ferma's big theorem, German mathematician kumer divided the prime numbers into regular prime numbers and non-regular prime numbers. In addition, he proved that ferma's big theorem was true for regular prime numbers, so as to achieve the first breakthrough proved by the ferma theorem. The regular prime number is defined by the bernuoli number: Set P> 3. If the bernuoli number B2, B3 ,..., Every molecule in the Bp-3 is not a multiple of P, so the prime number P is called a regular prime number, otherwise it is called a non-regular prime number. The following lists the numerator of several bernuoli numbers:

F8 3617

B18 43667

B20 174611

B22 854513

B24 236364091

Bernuoli numbers are widely used in mathematical analysis and approximate calculation.