By Peter Barlow

ISBN-10: 1429700467

ISBN-13: 9781429700467

Barlow P. An hassle-free research of the speculation of numbers (Cornell collage Library, 1811)(ISBN 1429700467)

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M Ä 1/Š D m 1 Ã 1 X tn . 2 Formulas for the Bernoulli Numbers Involving the Stirling Numbers A formula expressing the Bernoulli numbers as a finite sum involving the Stirling numbers of the second kind has been known at least since Kronecker2 [62]. (See also an account of von Staudt in Sect. 6 (6), the Stirling numbers can be expressed as a finite sum involving only the binomial coefficients. , involving only binomial coefficients and simple polynomials) double sum. Gould [36] studied various historical sources for analogous formulas, and he concluded that, at least until around 1970, the very fact that there are closed finite formulas (without involving infinite sums and integrals) for the Bernoulli numbers was not widely known.

T/ D 1 nD0 bn t . n ! ai bn i tn t n 1 i D1 C ! t/: 1 i D0 i tn 1 i D0 ! i ai bn 1 X nD1 i D0 1 n X X nD1 ! 0 i D0 ! n i /bn i tn 1 20 1 Bernoulli Numbers Also, when R Q, the integral Z tX 1 an t n dt is defined formally by term-by-term 0 nD0 integration: Z tX 1 an t dt D n 0 nD0 1 X an nD0 t nC1 t2 t3 t4 D a0 t C a1 C a2 C a3 C nC1 2 3 4 : In order to treat the generating function of Bernoulli numbers, it is convenient to generalize formal power series to Laurent6 series (with a finite number of terms with negative powers).

6 (6) on p. 30 we have ! 3 X n l 3 ln . 1/ . 1/ mŠ D l m m lD0 D0 3 C 3 2n Á1 . a b/, then we say that a is congruent to b modulo n, and we write a Á b mod n. ) This shows that . m C 1/ is an integer. We have thus shown that if m C 1 is a composite number, . m C 1/ is an integer. Next, suppose that m C 1 equals a prime number p. 6 (6) that ! p 1 X n 1 n l p . 1/ mŠ . 1/ D l m l m lD0 Á p 1 X l n mod p (since 1 mod p 0 mod p if p if p p 1 l Á . 1/l mod p) lD0 Á 1 divides n, 1 does not divide n. 1) The last congruence can be seen as follows.

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An elementary investigation of the theory of numbers by Peter Barlow

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