AN ARITHMETICAL THEORY OF THE BERNOULLI NUMBERS...Lucas of the ultra-Bernoulli numbers(6). Bernoulli...

30
AN ARITHMETICAL THEORY OF THE BERNOULLI NUMBERS BY H. S. VANDIVER In the present paper we shall describe a method which enables us to find many new types of relations concerning the Bernoulli and allied numbers. The scheme might be described as ultra-arithmetical in character. It depends mainly on the following idea. Let a and b be rational with a = &, modulo p, where p is any prime integer. If a and b do not depend on p, it then follows, since there is an infinity of primes, that a = *>. A similar method has been employed in other parts of mathematics; for example, HasseO) in a paper on algebraic geometry uses the method and com- ments upon the success it has had in various lines. Perhaps the simplest looking formula in which a Bernoulli number ap- pears alone on one side of the relation is as follows, if Sn(p) = 1"+ • • • +(p —l)n, -■ ô„ (mod p), P where «+1 <p, in which case, of course, the left-hand member of the con- gruence is an integer. In order to take advantage of this simplicity we employ extensively the function which we have called in a previous paper the Mirimanoff polynomial(2), namely, the relation (1) which follows. This is connected with the previous congruence, if we note that £\t) = Sn(P). In general we employ more or less obvious identities involving one or more indeterminates, then operate thereon, using the method of formal exponential differentiation explained in another paper(3). The elementary function from which the Mirimanoff polynomials are generated by this process is xm — 1 _ x - 1 Presented to the Society, September 5, 1941; received by the editors April 5, 1941. (1) Abhandlungen Göttingen, vol. 18 (1937), pp. 51-55; cf. also Vandiver, Bulletin of the American Mathematical Society, vol. 31 (1925), p. 348; in particular, the proof of II. There is a misprint in the first congruence involving H. The right-hand member should read ac in lieu of ae. (2) Vandiver, Duke Mathematical Journal, vol. 3 (1937), p. 570; so-called because Mirimanoff, it appears, first investigated its properties extensively in an article in Crelle's Journal, vol. 128 (1905), pp. 45-68. (3) Vandiver, On formal exponential differentiation in rings, Proceedings of the National Academy of Sciences, vol. 28 (1942), pp. 24-27. 502 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Transcript of AN ARITHMETICAL THEORY OF THE BERNOULLI NUMBERS...Lucas of the ultra-Bernoulli numbers(6). Bernoulli...

Page 1: AN ARITHMETICAL THEORY OF THE BERNOULLI NUMBERS...Lucas of the ultra-Bernoulli numbers(6). Bernoulli numbers of the first order are considered in 3 to 11 of this paper. In 5 a generalization

AN ARITHMETICAL THEORY OF THEBERNOULLI NUMBERS

BY

H. S. VANDIVER

In the present paper we shall describe a method which enables us to find

many new types of relations concerning the Bernoulli and allied numbers. The

scheme might be described as ultra-arithmetical in character. It depends

mainly on the following idea. Let a and b be rational with a = &, modulo p,

where p is any prime integer. If a and b do not depend on p, it then follows,

since there is an infinity of primes, that a = *>.

A similar method has been employed in other parts of mathematics; for

example, HasseO) in a paper on algebraic geometry uses the method and com-

ments upon the success it has had in various lines.

Perhaps the simplest looking formula in which a Bernoulli number ap-

pears alone on one side of the relation is as follows, if Sn(p) = 1"+ • • • +(p — l)n,

-■ ô„ (mod p),P

where «+1 <p, in which case, of course, the left-hand member of the con-

gruence is an integer. In order to take advantage of this simplicity we employ

extensively the function which we have called in a previous paper the

Mirimanoff polynomial(2), namely, the relation (1) which follows. This is

connected with the previous congruence, if we note that

£\t) = Sn(P).

In general we employ more or less obvious identities involving one or more

indeterminates, then operate thereon, using the method of formal exponential

differentiation explained in another paper(3). The elementary function from

which the Mirimanoff polynomials are generated by this process is

xm — 1

_ x - 1

Presented to the Society, September 5, 1941; received by the editors April 5, 1941.

(1) Abhandlungen Göttingen, vol. 18 (1937), pp. 51-55; cf. also Vandiver, Bulletin of the

American Mathematical Society, vol. 31 (1925), p. 348; in particular, the proof of II. There is a

misprint in the first congruence involving H. The right-hand member should read ac in lieu of ae.

(2) Vandiver, Duke Mathematical Journal, vol. 3 (1937), p. 570; so-called because

Mirimanoff, it appears, first investigated its properties extensively in an article in Crelle's

Journal, vol. 128 (1905), pp. 45-68.(3) Vandiver, On formal exponential differentiation in rings, Proceedings of the National

Academy of Sciences, vol. 28 (1942), pp. 24-27.

502

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THE BERNOULLI NUMBERS 503

In view of the congruence (68) below it seems to me that the theory of

Mirimanoff polynomials has often been obscured by the theory of the Euler

polynomials defined in (12) and (13). I think that this is unfortunate, as the

Mirimanoff polynomial has a much simpler algebraic form. Each type is gen-

eralized in §2 of this paper. In (19) instead of a congruence we derive an equa-

tion involving the two types of polynomials.

If in place of the simple congruence involving S„(p) given above we em-

ploy a known congruence such as the following:

(»* - l)bi-= yaa,_1 (mod p),

i

where ya = —a/p (mod «), we are forced to use an extension of the Mirimanoff

polynomials and relations involving the number

(m* - l)bn

n

in lieu of bn itself. The former number has appeared in a great number of in-

vestigations concerning Bernoulli numbers, but its properties seem to be

quite different in many connections; for example, I quote Frobenius as fol-

lows: "... die Tangentenkoefficienten deren Theorie man in den bisherigen

Darstellungen nicht scharf genug von der eigentlichen Bernoullischen

Zahlen geschieden hat."

In a previous paper(4) the writer introduced the idea of Bernoulli numbers

of various orders such that we call bn(m, k) = (mb + k)n, m-AO, a generalized

Bernoulli number of the first order; and a number of the form, for r>l,

(mrbM + Wr-ii''-" + • • • + nixb' + w0)n = bn(mr, mT_x, • ' ' ,m0),

where this expression is expanded in full by the multinomial theorem and ba

substituted for &„", t = l, 2, • • • , r, and where the m's are integers, mt9*0,

a Bernoulli number of the rth order. This is an extension of the definition of

Lucas of the ultra-Bernoulli numbers(6).

Bernoulli numbers of the first order are considered in §§3 to 11 of this

paper. In §5 a generalization of the von Staudt-Clausen theorem is derived

which applies to Bernoulli numbers of the first order (Theorem I). In another

(4) Vandiver, Proceedings of the National Academy of Sciences, vol. 23 (1937), p. 555.

(6) Frobenius, Berlin Sitzungsberichte, 1910, p. 810. "Die Bezeichnung der Werte

(b + b')", • • • , (b + é<« + ¿m H-+ i<»>)»,

als Bernoullische Zahlen höherer Ordnung oder gar als ultra-bernoullische Zahlen scheint mir

wenig glücklich gewählt und mehr von abschreckender Wirkung zu sein."

The writer differs from Frobenius regarding this. These numbers, as well as the generaliza-

tions of them considered in this paper are shown to be natural analogues of the ordinary

Bernoulli numbers.

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504 H. S. VANDIVER [May

paper(6) a generalization of certain congruence properties of the numerators

of Bernoulli numbers is obtained which applies to the numerators of any

Bernoulli number of the first order. It is noteworthy that these generalizations

are very little more complicated in statement than those theorems which

apply to the ordinary Bernoulli numbers.

In order to illustrate the varied applications of our method and to de-

velop a connected theory, we give new proofs of several known theorems.

In particular, although Theorem I was proved in previous papers(6), we give

two new proofs of the same, as the result and the material in the proofs are

both important in our theory.

The properties of the Bernoulli numbers of the second order are considered

in §11 to §15. It seems to me that from many standpoints this type of

Bernoulli number is the most remarkable. For example, we note from Theo-

rem III that the only prime factors occurring in the denominator of such a

number, say (kb+jb' + h)n, «even, are divisors of jk and are also found among

the von Staudt-Clausen primes of order n. As noted in §16, this property does

not carry over to Bernoulli numbers of higher order. Also (Theorem IV,

Corollary I), any Bernoulli number of the above type can be expressed as a

linear combination of Bernoulli numbers of the first order with coefficients

whose denominators divide the integers k and/. In particular cases the equiv-

alent of this result may be expressed in terms of the roots of unity (Theorem

V). Results of an entirely new type are given in Corollary I and Corollary II

to Theorem V.

Since congruence methods are employed throughout this paper, one might

imagine that many new congruences concerning the Bernoulli numbers could

be obtained aside from those given here. Such is indeed the case, but their

statement will be reserved for other papers.

1. Euler and Mirimanoff polynomials. We write the Mirimanoff polyno-

mials in the form

(m) , . a a a 2 , , a m—1

(1) /„• (x) = 0 + 1 x + 2 x + ■ ■ ■ + (m - 1) x ,

where 0°= 1.

We shall first show that, for « > 0,

(2) x[(f(x) + I)" m Mx) (mod p),

where fi(x)=f¡p\x), p is prime, and the left bracket symbol in the left-hand

member signifies that the wth power of (f(x) + l) is to be taken symbolically,

that is, after development by the binomial theorem the exponents are de-

(6) Vandiver, On simple explicit expressions for generalized Bernoulli numbers of the first

order, Duke Mathematical Journal, vol. 8 (1941), pp. 575-584; Carlitz, Generalized Bernoulli

and Euler numbers, ibid., pp. 586-587.

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1942] THE BERNOULLI NUMBERS 505

graded to subscripts and, in particular, the last term is expressed asf0(x). Also

for w = 0 any such expression is taken as unity.

Consider

(3) x(l + x + x2 -\-+ x"-1) = x + x2 + ■ ■ ■ + x".

Now employ formal exponential differentiation (Vandiver(3)).Differentiating

this expression « times with respect to x, we obtain

*[(/(*) + Y)n = x + 2nx2 + ■ ■ ■ + (p - l)nxP~l + pnxp

(4)= fn(x) + p*x*,

from which (2) follows.

Now consider

(5) x + x2 + ■ ■ ■ + x" = (1 - x")x/(l - x).

Differentiate this « times, where ¿denotes exponential differentiation ; we find

r¿~/ * M pF(x)x + 2"x2 + ■ ■ ■ + p"x" = (1 - x»)\-1-J + —-,

F \_dx»\\-x)\ (1 - *)»

pF(x)

or

V dn / x \-| pF(x)fn(x) + p"x" = (1 - x')\—(.-- +7T-r

ldxn\l — x/J (1 — x)n

Yd"/ x \1 G(x)Mx) = (1 - *») — (--) + p \

Ldx"\l - x/J (1 - x)n(6)

(1 - x)'

We then have, for n > 0,

(!)

where we write M(x)/N(x)=0 (mod p), if M(x) and N(x) are polynomials in

which each coefficient of M(x) is divisible by p, while not all the coefficients

of N(x) are so divisible.

We write

1 — xY dn / x \~|Hn(x)= - —(--) L

x Ldxn\l — x/Jso that

(1 — xp)x(8) /,(*) = (- 1)" —-Hn(x) (mod p).

1 — X

Now consider

xr\l + x + x"-+ • ■ ■ + xp~l) = 1 + x + x2 + ■ ■ ■ + xp-1 — x"-1 + x~p ■ x*"1.

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506 H. S. VANDIVER [May

Differentiating « times, we obtain

1 .— l(/(z) - 1)" = fn(x) + (p — i)nx"-1x~1' — (p — l)nxp-1 (mod p),X

whence

(9) [(f(x) - 1)" - */.(*) + (- 1)»(1 - x») (mod p).

We now express each/0(x) in this relation in terms of Ha(x) by using (8), and

this gives(7), modulo p,

(10)

or

1 - x" r 1 -*-- (-Ü - l)"-(- 1)»+ (- I)»—-

1 — x L 1 —

x"

X

(- l)"x2-— #„ + (- 1)«(1 - x'),1 — X

(11)

r i - x" .. . i - x"~\- 1)» x--{[(77+D--1} +--

LI — x 1 — x A

r i — x? I^ (- l)"\x2——-Hn+ (1 - **)J.

Divide (11) by ( — 1)" and (1— x") and multiply by 1—a: to get

x[(H + 1)" - x + 1 = x2Hn + 1 - x (mod p),

or

x[(H+ 1)» m x2Hn (mod p),

whence

[(H + 1)" = xHn (mod/>).

But this relation is independent of p, an arbitrary prime, hence is an equation,

and we have(8)

(12) [(H + 1)» - xB», «>0.

Taking 770= 1, we obtain,

üi = l/(x - 1),

Ü2 = (1 + *)/(* - l)2,

(7) In (10) we substitute the Ha terms ( — 1)" of the (— H — 1)" expansion and add

/o = (l — xp)/(l —x)( — l)n of the (/+1)B expansion, because (8) does not hold for n = 0.

(8) Frobenius, Berlin Sitzungsberichte, 1910, p. 828.

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1942] THE BERNOULLI NUMBERS 507

773= (1 + 4x + x2)/(x - l)3,

Hi = (i + il* + il*2 + x3)/(x - iy,

Hi = (1 + 26* + 66*2 + 26x3 + x*)/(x - l)5.

Now write

(13) 77n = Rn(x)/(x - 1)».

The Rn(x) are the Euler polynomials,

Ri= 1,

Ri = 1 + x,

Putting (8) in (2) we obtain

x2-— {[(- 77+1)"- 1} + x-— = (- 1)»*-X—Hn(m°àp),1 — x 1 — x 1 — X

x2{ [(- H+ 1)» - 1} + * « (- l)nxHn (mod p),

(14) x2{ [(H - 1)« - (- 1)»} + (- l)»x = xHn,

x[(H - 1)"- x(-l)»+(- 1)" = Hn(modp),

x[(H - 1)» - *(- 1)" + (- 1)" = Hn,

which is another recursion formula for the 77's.

We also have

[(k + (H+ 1)") = ¿" + Cn,xk"-l(H + 1)

(15) + Cn,ik»-2(H + I)2 + ■ ■ ■ + (H + 1)»

= kn + Cn.xk^xHx + Cn,2kn~2xH2 + • • • + XHn,

and

(16) [(k + H)"= kn + Cn.lk^Hx + Cn,2k»-2H2 + ■■■ +H„.

Multiply (16) by x and subtract from (15). We then obtain(8)

(17) [(k + H + 1)" - x[(k + H)n = k"(l - x).

Setting k = 0, 1, 2, ■ ■ • , r in (9), we obtain

[(77 + 1)» - x77n = 0,

[(H + 2)» - x[(H + 1)« = 1"(1 - x),

[(II + 3)" - x[(H + 2)" = 2"(1 - x),

[(77 + r)» - x[(H + r - 1)" = (r - 1)"(1 - *).

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508 H. S. VANDIVER [May

Now multiply the first of these by xr~l, the second by x'~2, and so on, and add.

We then obtain

[(H + r)" - x'Hn = (1 - x) [l"xr~2 + 2nxr~3 + 3"xT~i + ■ ■ •

+ (r-2Yx+(r- 1)»](18)

= (1 — x)x /»[—J-

Using a similar scheme, we obtain(9) from (14)

(19) /+1[(ü - r)n - xHn = (- l)"(x - l)fn+1\x).

This is a rather curious relation ; the expression on the left involves quotients of

polynomials, while /„(%) is a polynomial for each r.

2. Generalization of the Mirimanoff and Euler polynomials and a related

formula. Set

fa(x, m, k) = k" + (k + m)ax + (k + 2m)ax2 + • • •

+ (k + (p - l)«)«**-1.

We now consider

ykn _ y">pxp)(21) -—- = yk + y(k+m)x _(_..._)_ yWH)«)^-!,

1 — ymx

Differentiate this expression a times exponentially with respect to y and set

y = 1. We then obtain (20) on the right; on the left we differentiate in the form

(22)ykil _ ympxp)

1 — ymx

and we then obtain

= (1 - y*»»*») ( —^-Y\1 - ymx/

(23) fn(x, m, k) m (1 - x») [—- (-—^-)1 (mod p).Ldyn\l - xym/X=i

We now set

Y~dn ( yk \"1 Hn(x, m, k)

(24) kHr^—) -(-!)■•Ldy" \1 — xym/jy=i 1 — x

We now seek a recursion formula for this function. We have

y—m

- (yk _L Xyk+m _|_ . . . _|_ (XP— lyk+{p— D""))

(25) i= - yk-m _J_ (yk _|_ Xyk+m _|_ . . . _J_ xp-lyk+(p-l) m\ _ xp-lyk+(.p-l) m_

X

(') This result war, obtained by Dr. A. M. Mood, following a suggestion by the writer.

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1942] THE BERNOULLI NUMBERS 509

Differentiate this relation « times exponentially and put y = l. We obtain

(26) [(f(x, m, k) - m)n = xfn(x, m, k) + (k - m)"(l — x") (mod p).

Now (24) with (23) gives

1 — xp _(27) fn(x, m, k) = (- 1)"-Hn(x, m, k) (mod p),

1 — x

which, applied to (26), gives

(28) [(77 + my = xHn + (m- k)"(l - x).

The first three TPs given by this recursion formula are

_ (m — k)(x — 1) + m k(l — x) + mxHx =-= —-1

x — 1 x — 1

_ \(m — k)(x — 1) + m]2 + m2xH2 =->

(x- l)2

[(m — k)(x — 1) + m]3 + m2x2(4m — 3k) + m2x(m + 3k)

(x - l)3#3 =

Now (28) is not a direct extension of (12). To obtain such an extension of

the latter formula we use

y*+mx(l — ypmxp)yk _|_ xyk+m _|_ . . . I xp— Xyk+(p— 1) m — Vk + xPy^V1"" = ->

1 — xym

which, differentiated » times with y = 1, gives

Yd" / yk+m M(29) /„(x, m, k) - k"(l - x") m x(í - x») - -) (mod p).

Ldyn\í — xym/Jy^x

Putting

Yd~»( yk+m M(30) Hn(x,m, k) = (- 1)-(1- *) —-) ,

Ldyn\l — xym/Jy=x

we then obtain

1 — xp(31) - /„(x, m, k) - kn(l - xp) = (- l)nx-Hn(x, m, k) (mod p).

1 — x

Using this in connection with (26), we find

(32) [(H+ m)n = xHn +(- k)"(l - x).

For & = 0, m = l, this reduces to (12).

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510 H. S. VANDIVER [May

3. Bernoulli numbers. Set

Snip) = 0" + 1" + 2» + ■ ■ ■ +(p - 1)"; 0° = 1,

where p is a prime. If n<p— 1, it is known that Sn(p)=0 (mod p). We then

write under this restriction

(33) Sn(p) - pan (mod ¿2),

where a„ is some integer.

Consider

(34) (1 + x + x2 + ■ ■ ■ + xr-^x = x + x2 + - - ■ + X*.

We now differentiate exponentially with respect to x, using Leibnitz's theo-

rem. We obtain

n

(35) £ Cn.ixfi(x) = x + 2"x2 + 3"x3 -\-+ pnx".,=0

Restrict « to be greater than 1. Now (35) can be written

n

(36) zZCn,iXfi(x) = fn (x)+pnx".i=0

Setting x = 1, we have

(37) Ê C„,iSi(p) = Snip) + P",i-0

which, in view of (33) and of the fact that «>1, is then

(38) ¿Z Cn.ipai = pan (mod p2).,=o

Dividing through by p, we may write in symbolic form

(39) [(a + 1)" = a„ (mod p).

Now consider the recursion formula

(40) [ib +1)*= bn

for n<p — 1. If we determine £>i, b2, ■ ■ ■ , bn-i in turn with the use of this, each

denominator of the fractions so obtained will be prime to p. In view of (39)

we may also obtain an in the same manner. Consequently, we may write

(41) Snip) =- Pbn (mod ¿2).

It is known that Snip) =0 (mod p2) if « is odd,greater than 1, and less than

p — 1. Consequently,

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1942] THE BERNOULLI NUMBERS 511

pbn = 0 (mod p2), bn = 0 (mod p)

for any p > « +1 ; hence.

¿>„ = 0, for » odd and greater than 1.

4. We have, obviously,

X" — 1 x(.P+k) _ I Xk — 1

(42) --** =--,x— 1 x— 1 x— 1

which may be written

(1 + x + x2 + • ■ • + xí'-'Ox* = 1 + X + • • • + x"-1(43)

+ x"(l + x + • • • + x*"1) - (1 + x + • • • + X*"1).

Differentiating this relation exponentially « times with respect to x and col-

lecting the terms whose coefficients are divisible by p2, we obtain

(44) [(/(x) + k)n = Mx) + xVfn\x) + npxVfn-x(x) + p'p(x) - fT(x),

where P(x) is a polynomial in x with integral coefficients. Put x= 1 ; we then

obtain

(45) [(S(p) + A)" = Sn(p) + npSn-i(k) + p2W,

where W is an integer. We employ (41) in connection with (45) for each 5

occurring in the expansion of the left-hand member and for each 5 occurring

in the right-hand member; after dividing the resulting expression through

by p, we obtain the relation

(46) [(b + *)» = b„ + nSn-i(k) (mod p).

As none of the terms in this congruence depends on p, we obtain the Bernoulli

summation formula

(47) [(b + k)" - bn = nS„-x(k).

With (46) as a base we shall now prove the formula^-10)

s-l

(48) [(mb + sm + k)n — [(mb + k)n — nm22 (k + im)n~l.i-0

Consider

(1 + xm + ■ ■ ■ + xm<-p-l))xsm+k — (1 + xm + ■ ■ ■ + xm^~l))xk

(49) = x^x* + xk+m + • ■ • + x*+(s_1)m)

— (xk + xk+m + ■ • • + xk+u~1)m),

(I0) Due to Glaisher, Quarterly Journal of Mathematics, vol. 31 (1900), pp. 193-199. For

another proof by the writer see American Mathematical Monthly, vol. 36 (1929), pp. 36-37.

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512 H. S. VANDIVER [May

where the k and m can be in any ring including the rational field. Employing

the concept of generalized exponents used by the writer (Vandiver(3)) the

exponential differentiation with respect to x gives

[(/(*») + ism + k)Y - [(/(*") + *)"

(50) = npmxpmik"-lxk + (A + m)n-lxk+m + • ■ •

+ ik + is - l)mn-l)x™) + p2Pix),

from which we easily obtain (48), after using (41).

Various arithmetical results not involving explicitly the Bernoulli num-

bers may be derived by the exponential methods we have been employing.

For example, if a is an integer such that

ad = 1 (mod p"),

then

n

(51) ixa" - 1)" = ¿C.#**(- 1)V.t=0

Differentiating exponentially, with respect to x, k times for k<m, we find

n

(52) Pix)ixad - 1)"-* = Y,C%.txiihi- iyaid,,-o

where P is a polynomial in x. Put x = 1. Then we obtain

(53) Pia* - I)-* = ¿ Cn,iiki- 1)««",•=o

whence

n

(54) £ C„,<i*(- 1)^" * 0 (mod />«(»-*))i-O

for /;<«. Putting

d = p — 1, a = 1, » = 1,

we have Fermat's theorem.

5. Generalized Bernoulli numbers of the first order. In (48) set s = p, a

prime. Then expand the left-hand member according to powers of p. The re-

sult can be expressed in the form

T P Y P2™Sn-i(m, k, p) = C„,i (mb + k)"-1 — + Cn.2 (mb + k)"~2 ̂ — + • • •

L « L «(55)

prmv-l r pn

+ - Cn-l,rp (Mb + k)"-l-T + ■ ■ ■ + — •r + 1 L «

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1942] THE BERNOULLI NUMBERS 513

From this we may show that for p>2 and (mb + k)° = l

(56) Sn(m, k, p) = [(mb + k)"p (mod p).

To obtain a proof by induction assume that in (55) for i<» — l,[(mb + k){ may

be expressed as a fraction with the denominator prime to p. Also for r>0

and p odd we have

(57) -— = 0 (mod p),r + 1

since pr>r+l for p odd, as is seen from

PT Ê; (1 + 2)r = 1 + 2r.

Hence (55) gives (56). For p even and w = 2, (56) also holds, since we may

verify that

(58) 2(mb + k)2 = Si(m, k, 2) (mod 2).

For brevity set

h„ = [(mb + k)\

Now for n odd, p odd, and m prime to p, the expression on the left of (56)

is divisible by p, for « is not a multiple of p—1 since p—1 is even, and for

m=0 (mod p), Sn(m, k, p) is obviously =0 (mod p). Hence h„ does not have p

as a factor of its denominator, except possibly when p = 2.

Now for p = 2 (57) holds for r>\, so that

2' = (1 + I)" > 1 + r

for r>l. Hence (55) gives for p = 2, «>1 and odd,

JL m22(sm + ky = 2hn +-C„+1,2Ä„-i22 (mod 2),

(59) 8_o » + 1

kn + (m + k)n = 2hn + mn(2hn-x) (mod 2).

Since kn = k (mod 2), for m odd we have

(60) l = 2hn+ (2hn-x) (mod 2),

and for m odd, « even, we have

(60a) 1 = 2hn (mod 2).

Now from this we cannot have both hn and hn-x with 2 in the denominator

for w odd. Also for m even 2 will not appear in the denominator for « either

odd or even. Now since 2 is in the denominator of h2 for m odd by (58), h-¡ is

integral from (60). Hence (60) and (60a) give hn integral for « odd and greater

than 1.

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514 H. S. VANDIVER [May

Consequently,

[(bm + k)n

is integral except when « = 1 with m odd, and also

(61) Sn(m, k,2) = hn-2 (mod 2)

for « even.

Consider

(62) hn-lZ3^^'p

where the p's range over all the distinct primes less than or equal to « +1. For

a particular p, say p', of this type the fraction

Sn(m, k, p')bn-

P'

may be expressed with a denominator prime to p' by (56) and (61), and the

remainder of the expression in (62) obviously has the same property. Hence

(62) must have no primes in its denominator and is therefore an integer. Also,

Sn(m, k, p) = 0 (mod p)

for m prime to p and «f^O (mod p—1), and obviously also holds for m=0

(mod p). For « = 0 (mod p — 1) and m prime to p

Sn(m, k,p)mp-lm-l (mod p),

whence the theorem follows(u).

Theorem I. If m and k are integers, then if « is even and greater than 0,

[! 1(mb + k)" = An - £ —.

<-i pi

where pi, p2, • ■ • , p, are the distinct primes which are prime to m and such that

w = 0 (mod pi — 1), An being some integer. If n is odd, then [(mb + k)n is an in-

teger except when « = 1 with m odd.

6. In another paper(12) the writer obtained the formulas

(u) This theorem was first stated by the writer without proof in Proceedings of the National

Academy of Sciences, vol. 23 (1937), p. 556, and the present proof was there briefly indicated.

Another proof was given in Duke Mathematical Journal, loc. cit., Theorem III. A third proof

is given in §7 of the present paper, but the ordinary von Staudt-Clausen theorem is assumed

therein.

(») Vandiver, Annals of Mathematics, (2), vol. 27 (1926), p. 174 (10); p. 175 (13).

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1942] THE BERNOULLI NUMBERS 515

o-l

(Xm - 1) J2 Ca,nk"-nmn+1bnfa-n(x)n=0

_ x(pk - l)(xp - !)(*- - l)Fa(p)

(63)p(x - pk)(x - 1)

1 p halx (x)— axm(xm — 1)22 22 —"-(mod p),

p i_0 x — p*

andmMx(p)

(64) Fn(p) = (n - l)p -^-^ + p2C(p).pp - 1

Applying the second formula to the first, dividing the first through by x and

setting x = 0, we find

(— l)n+lpkf _i(o) k~1

(65) Um, k) m ^' -- P J* + Z Z ¿""V"* (mod p),p 1 — Pp 1=0 /,

where ^p' indicates summation over all the rath roots of unity except unity,

and 22p indicates summation over all the distinct roots of unity; and where

[(mb + ky - bn(66) -= t„(m, k).

n

When k^m, (m, p) = l,m>l, the relation (65) may also be given in the form

, LN ^, (- D"+1Pfc/n-l(p) [*,£> v w(67) t„(m, k) m }2 -■ + m 22 (¿ ~ sm)"-1 (mod p).

1 - Pp «_i

This is a companion formula to (47), but the latter is not obtainable(13) from

(67), as (67) does not hold for re = l. Using (8) and (13), we find

(68) fn(x) = x(l - x)"-"-1Pn(x) (mod p).

Applying this to (67), we find the equality

^ (- i)"+V+1P„-i(p) [*£?](69) tn(m, k) = 22--;-:-— + f»ZC- ww)""1-

, (l - p)" .-i

In the relation (65) suppose that m>k. Then the last term on the right

becomes zero, so that we obtain(,4)

, ., (- l)»+lpt+lPn-l(p) p*+lP„_l(p)

(70) tn(m, k) = 22'- - £'-7— ■(1 - p)" , (p - 1)"

(l3) Relations (65) and (67) were given without proof in another paper by the writer,

Proceedings of the National Academy of Sciences, vol. 25 (1939), p. 200.

(u) Frobenius, loc. cit., p. 827.

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516 H. S. VANDIVER [May

7. Application of some relations due to Frobenius(14). We have

m "ILl1

(71) -:=£*>',p—l t-i

and from (69)

^ / m Vmnn(m, k) = - zZ'i—— ) Pk+lRn-i(p) + I

(72)/ m—1 \ n

- Z'(^Z*p8Jp*+1*»-i(p) + 7

where 7 is an integer. The expression on the right is a polynomial in p with

integral coefficients and when summed over all roots of unity except 1 will

be an integer, as p+p2+ ■ ■ ■ +pm~1= —1. Hence mntn(m, k) is an integer.

To show that m((mb + k)n — b„) is an integer we note that

mn

- [[(mb + k)n - bn]n

is an integer; expanding this, we obtain

mn r mn

(73) - [(w" - l)bn+ kn + m(R)] = -K.n n

Since (73) is an integer, the fractions with denominators prime to m will can-

cel out and we shall be concerned only with primes in the denominators which

divide m. By the von Staudt-Clausen theorem these appear only to the first

power, hence in (73) m(R) is an integer.

Let d denote the denominator of bn; then

m'

(74) - (K)«

is an integer in which m' = (mn, nd). But mn is divisible by m', since prime

factors of d occur only to the first power; hence we can replace m' by mn in

(74) and obtain the desired result.

We shall now show that [(mb + k)n is an integer for « odd and greater than

1 as in Theorem I. First note that m[(mb + k)n is an integer, since bn is zero

for « odd and greater than 1. Now if [(mb + k)n were a fraction, it would

have a denominator which divided m. Hence, since every term of [(mb + k)n

contains m except the final term kn, we see that [(mb + k)n is an integer,

except in the case when «=1 and m is odd. In the latter case we have

mb + k=-(l/2)m + k.

8. We shall now give another proof of our Theorem I, using the von

Staudt-Clausen theorem in the expansion of [imb + k)n, that is,

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1942] THE BERNOULLI NUMBERS 517

(75) [(mb + k)n = m"b„ + nmn~lbn-xk + ■ ■ ■ + k",

and obtaining

(76) (mb + ky = i + Y! —L r,

where the mt are integers less than r,-. Now

(77) m[[(mb + k)* - b„] = 7;,

where 7, represents an integer. Hence,

(78) ml + m 2^-m«. + »iL — — l\,ri qt

or

(79) m(z=+z-)-/.where 72 represents an integer which combines ml, ma„, and terms wherein qt

and r, divide w. Now in (79) no qt or r, divides w, hence

(Z=+Z-)must be an integer 73; therefore no r, is different from some qt and vice versa,

and hence

22*^ = 1,It

Since each term in this sum must be an integer and since | w,| <qt, we obtain

nti= —1. Thus, we have proved Theorem I.

9. Illustration of our congruence methods. In order to illustrate our con-

gruence methods further we show how a known property of the generalized

è's may be easily derived. By direct expansion we find that

t>—1 n

2~2 (k + smy = 2~2 Cn.iSi(P)mk'~i.«=0 «=0

Now let «+1 <p; using (41) we find that

r 7.-1

(80) p\(mb+ k)n = 22 (k + sm)n (mod P2),

whence

m— 1 T~ w—\ p—1 my—I

P22 \(m0 + 0" = ¿2 ¿2 (ms + 0" = 12 i" = S„(mp) m mpbn (mod p2).i-O L i=0 s=0 t=0

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518 H. S. VANDIVER [May

Consequently (16),

m-l

(81) E [imb + 0" = mb*,t=0

10. We now develop some congruences of a novel character involving the

Mirimanoff polynomials, and leading to congruences of a new type relating

to the Bernoulli numbers. Obviously,

XP — \ xk _ J

(**- 1) —- = (*» - 1) •--,x — 1 x — 1

whence, for p an odd prime,

(** - l)fv-i(x) - kxkf^i(x) + k2xkfp-3(x)-+ xkk^fo(x)

t-i■ (x" — l)zZlv~lxl (mod p),

or, if [u] is the greatest integer in u,

- kfp-i(x) + k2f^3(x) + ■■■ + **-"/.(*)

fc—1 % _ /pP

m (X' - 1)Z lp-xxl-k + (x~k - 1) -(mod p)¡=i 1 — x

fxp-k _ x-k}<x _ x*) ix-k _ tyrx _ xp)

1 — X 1 — X

- (x? - l)(x"~k + x2"~k + ■ ■ ■ + xPlUpl-k) (mod p),

m 1 — x*-k - (xp — l)(x"~k + x2p~k + • • • + ^p[*/pl-*) (mod p),

where we understand that the second member is zero if [k/p] = 0. Assume

p>n+l, replace k by kr and multiply by rv~x~n, let r take on 1,F2, • • • ,p— 1,

and add. We have, after dividing by (l—xp)

(- l)-+»*-/^-i(») *£ ! - xP~kr , s ..-= £ —-— ir)r-i-»

1 — X" r=l 1 — X"

+ V* j.p-1—n/^p—tr _|_ . . . _|_ xp[kr/p]-kr\

(82) a:P/p-n-i(a:-*)

1 — x"

p-i+ S\. rp~l~n(xv~kT + • • • + XPlkr/p]-kr\

r=l

(l5) Kummer, Crelle's Journal, vol. 40 (1850), pp. 119-121; Blissard, Quarterly Journal of

Mathematics, vol. 4 (1861), p. 288. There are also later references.

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1942] THE BERNOULLI NUMBERS 519

Let x = p, an wth root of unity different from 1, (m, p) = 1 and let 22' indicate

summation over all p's9*l, and set k = 2. We obtain, modulo p,

(- »-«/„-(p)*- _

1 — pv

(83)- 22'f" + 2Z' E ^-v-*.

1 — PP «=(p+l)/2

Let m<(p — l)/2 and (wi, 2) = 1, p>2 and m > 2, and consider the expression

p-i22, sp~1-npp-*k,

«-(P+D/2

and we shall determine the terms in which the exponents of p are =0 (mod m).

If I is one such, then

p — Ik = 0 (mod m),

where I is in the set (p+l)/2, ■ ■ • , p — 1. From the above congruence

(p+m)/2 is a solution, but (p — m)/2, although it satisfies the congruence,

is not in the set mentioned. Hence the solutions we need for [p/m] odd are:

p + m p + 3m p + [p/m]m-> -> • - • ,-•

2 2 2

For [p/m] even, it is replaced by ([p/m] — 1) in the above.

p-i p-i » / p _|_ m\p-i-»(84) 22' E s>-i-».pp-*m- 22 sH"n + «IL--)

«=(P+l)/2 «-(p+D/2 v-1 \ 2 /

where h= [p/m] or [p/m] — l according as [p/m] is odd or even.

Nowp + jm jm-= — (mod/>),

so that the last term on the right is congruent, modulo p, to

Í/P + ÎW— (P + 3V-1-» /p + l\*-i-«\

where h is defined as above.

If in (83) we employ (67) where the k used in the latter congruence equals

zero, we obtain new relations involving Bernoulli numbers in view of (66).

11. Bernoulli numbers of the second order. We employ the identity(16)

(") Vandiver, Annals of Mathematics, (2), vol. 29 (1928), p. 171.

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520 H. S. VANDIVER [May

;'— I k-1

¿Z xnyk-a" £ y'*'-*'x' — yk „_o 1=0

(85)(* - l)(y - 1) y - 1 x- 1

where an= [nk/j], b¡= [lj/k], [u] is the greatest integer in u,j and k are posi-

tive integers, x and y arbitrary. Multiply by (xp— l)(yp — 1), where p is a

prime greater than «, set x = x2*, y = yz\ divide by zik, differentiate exponen-

tially a times with respect to z and set z = 1. We then have, reducing the right-

hand member, modulo p3, and where h^'k)(x) is the/a(a;, w, £) of (20),

(x - y ) (¿/(x) +//"(y))° = y (x" - 1)£ /y "**i ' (y)L n=0

+ 2^ a«x y />Aa-i (y)n=0

2 p *-a„^-, 2 (j,c„)

+ C„,2JS; x y 2^p ha-2 (y)

— x'(y" — i)zZy x 'h° ' (x)¿=0

I—1

P+l 1-bl .(.k.di), . T-1 P+l J-"! , («.«I), .

— 0-P] Z-,y X «a-1 (X)¡=0

- Co,27 /> £ Äa-2 ' (a:) (mod p ),

where cn denotes the least positive or zero residue of nk, modulo/, while d¡

denotes the least positive or zero residue of //, modulo k. Now set x = xzk and

differentiate once with respect to 2; we then have, modulo p3, after setting

z = l,

kjx'[(kj(x) +jf(y)Y + (x> - yk)[Dl[ikfix) + jfiy))aUi

k p= y (x i)[^(g,vv^r",(y))]zi

(87) , , P k ^-, n —o„ (),c„) ^-, 2 k-an (i,c„)

+ pkx y ¿~, x y na iy) + 2-, ak ip + n)y />A„-i (y)n—0 n=0

¡fe-1ZP+I J-»I , (k.di) .2 ^-, p+i )'-!>! (i.di) .

y x ha (x) — apj ¿^ (« ~ Oy x «0-1 (x)1=0 ;=0

plus terms of the form p2g(x), where each g is divisible by some h¡ix).

Setting x = y = l, dividing through by p2, and using

pirb + s)n = sn + (r + s)" + (2r + s)n + ■ ■ ■

(88)+ ((P- l)r + sY (mod p2),

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1942] THE BERNOULLI NUMBERS 521

which follows from (55) for p>n, we find, since the result is independent of p,

j-i r f-i

kj \(kb +jb'Y = k 22 \(jb + cn)a + a2~2 k2n \(jb + cn)<-1L n=0 L n=0 L

- ajJ2 \(kb + dd' - aj222 \ (kb + dty~\k - I),¡=o L ;=o L

or since by (81)

/-i r *_l i

(90)

we may write

Z \(jà + CnY = jba; 22 \(kb + dda = kba,n=0 L 1-, il L

*/[(** +jb')a = jk(l - a)ba - ak2j2ba-x + a22 k2n\(jb + i»)-1(91) L n=° Lv ' k-x r

+ a^/2/ (kb + dty-K

Forh = 2,j = l, wefind(17)

(92) 2 [(2* + b'Y = 2(1 - a)ba - 4aô0_i + a(2b + l)—1.

Now since ( — b)n = bn except for » = 1, we have

(93) [(** - jb'Y = [(kb + jb'Y + ajk*-'ba-x,

whence from (91)

kj\ (kb -jb'Y =jk(l - a)ba - ak2j2ba-i + ajk°-lba-x

(94) ,_l r *_1 r+ a22 k2n\ (jb + CnY~l + a22M (kb + diY'K

Setting/ = ^ = 1 in (91) and (94), we obtain the well known relations

(95) [(b + b'Y = (1 - a)ba - aba-x,

and

(96) [(b - b'Y = (1 - a)ba.

Now for a odd we have

(97) [(kb + jb'Y = akbx(jby-1 + a(kb)"-ljbx,

so that from (91) we have the formula

(l7) Bell, these Transactions, vol. 24 (1922), p. 106.

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522 H. S. VANDIVER [May

ak2bxba-xj" + aj2bxba-xka = abxba^x(k2ja + j2k")

j'-i

= — ak2i2b_i 4- c

(98)

' r= - ak2j2ba-x + a22 k2n\ (jb + c)-1

n=0 L

k-X p

+ a¿2pi \ (kb + d,y-\¡=o L

There is an analogous relation from (94).

12. If h is any integer, we have from (91)

|(*i +jb' + hy=J2 Cn.a \(kb + jb'Yk"-"

= jk22 Cn.abahn-a - jk22Cn.aabah"-a(i=0 o=0

n )-l

(99)

- k2j222aCn,ahn-aba-x + ¿ Z k2iaCn,a\(jb + aY-lh»-°a=X a=X t=0 L

n *—1 p

+ lEyiflC.,. (kb + diY-1o=l i-O L

= jk\(b + h)n - jkn22Cn-i.a-xhn-abaL a-X

71

- k2j2n22 Cn-X,a-xhn-°ba-Xo=l

+ ¿ 22 k2inCn-x.a-x\(jb + c<)-1*—o-=i >=i L

+ E 22plnCn-X.a-x\(kb + rf,)-1*—,o=l 1=0 L

where we have changed the notation employed in (91). w now takes the place

of a and c, denotes the least positive or zero residue of ik, modulo j. Now

(100) 22Cn-l,a-lbah"-° = Ïb(h + o)""1,a-i L

and

(101) [b(h + J)-1 = [(h + b)» - h[(h + by~\

and we have this theorem :

Theorem II. If h, j and k are integers, k>0, j>0; c< represents the least

positive or zero residue of ik (mod/) ; d¡ represents the least positive or zero residue

of Ij, (mod k); n>0; then

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1942] THE BERNOULLI NUMBERS 523

jk[ikb+jb' + h)n = jk[(b + hY - ßn[(b + h)"

+ jknh[(b + h)"-1 - k2j2n[(b + h)n~l

j-i r

(102) + k2nJ2 i\ (jb + Ci + h)"-1

t-i r-

+ j2n¿Zl\ (kb + d, + h)"-\i=o L

Now since ( — b)" = ba except for a = 1, we have

(103) [(kb - jb' + hY = [(kb + jb' + hY + nj[(kb + h)*~K

This formula with (102) gives the following theorem:

Theorem III. If h, j and k are integers, n even and greater than 0, then

[! 1(kb+jb' + h)n = I - (1 - n)jk}Z —'

i=i pi

where the p's are the distinct primes such that w=0 (mod pi — 1), I being some

integer.

This is an analogue of the generalized von Staudt-Clausen theorem (I).

Now for « odd, (102) and (103) give

Theorem IV. If (« + 1) is odd, then for n greater than zero

(105) jk \(kb + jb' + *)-+> = I1 + (n+ l)jk (— + — - h) ¿ -,L \ 2 2 / ¡_i pi

where 7i is some integer, k, j and h are integers, kj is odd and the p's are defined

as in Theorem III.

It is not clear that this result is an analogue of any theorem involving the

Bernoulli numbers of the first order.

We also have from (102) and (103) the

Corollary I. ¿4«y Bernoulli number of the second order can be expressed

as a linear combination of Bernoulli numbers of the first order with coefficients

whose denominators divide the integers occurring in the original number.

Corollary II. The expression

(106) jk[(kb + jb' + h)n + jk(n - 1) [(b + h)n

is an integer if « is even,j, k and h integers.

The relation (102) gives, employing Theorem II of another paper(18) the

congruence

(ls) Vandiver, Duke Mathematical Journal, loc. cit.

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524 H. S. VANDIVER [May

(107) jk[(kb +jb' + h)n = 0 (mod pa),

where w is odd, p prime, n — l=p"r, (r, p) = l, (p, />,•) = 1, pi being any von

Staudt-Clausen prime of order n — 1. If « is even, this relation also holds if

« =0 (mod p"), (p, q) = l, where q is any von Staudt-Clausen prime of order w.

13. We shall now prove that

(108) ¿>a_i(f - j") m2j2i \(jb + Ci)*~\•-o L

with a odd and c, denoting the least positive or zero residue of ik (mod j).

Assuming (108) true for all values of k such that k<j, (k, j) = l, we have

from (91)

- ba-x(kY + fk") + 2k2j2ba-x

(109) M= 2 22k2i\ (jb + c,)-1 + 2 22fl\ (kb + di)"-\

i=o L ;=o L

and also

(110) - ba„x(k2j° + j2k° - 2k2j2) = 22, k2i \(jb + aY~l + j2(k2 - k°)ba-x,¿-o L

using c, = cr wheredi = dr with r=l (mod k), (r,k) = l,r<k; whence, the result.

14. Let p' = l; then

(111) \(jb + cny =ba+(- 1Y+Ia22' pCnfa~l(p) (mod p), a > 1;L 1 - pp

where 22' indicates summation over all distinct/th roots of unity different

from 1 ; then

-ba(j2 - ja+1) = Z n \(jb + CnY

(112) -1 L

- £«*«+(- D-'ag 2Z'^^-^ (mod p),n-l n-1 1 — P"

or

(113) í^4.. (_,)-.„£Z,5!^W,2 n_l 1 — p"

(114) - = ¿J-' a even and greater than 1.2a ** (I - p*)(l - P")

The latter relation may be proved directly by noting that if (k,j) = l,

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1942] THE BERNOULLI NUMBERS 525

fa-iip) p-*(- O-V-iGr*)(115)

Hence

(1 - p«)il - p-) (p* - l)(p-» - 1)

(116) E'-^—=zZ' "pVo"i(p)

and

(1 - p*)(l - p") ~ (1 - p*)(l - p-)

/o-l(p) - Pkfa-lip)y, / /a-i(p) - p*/a-l(p) \

\(i - p*)(i - p") (i - p*)(i - p')/

/U(p) f,(l-i%. . .= 2Z -= (- 1)"- (mod p).1 — p" a

15. Take the obvious identity

x — y y x

ix - l)(y - 1) y - 1 x- 1

((85) reduces to this for k = l,j = l). We may then write

(118) (x - y)fo (*)/„ (y) = y(x - l)/0 (y) - x(y - l)/0 (x).

Set x = x2, y = y2, divide by z, then differentiate each member exponentially

a times with respect to z, and reduce the terms, modulo p2; after setting 2 = 1,

we find

(x - y)[(f(x) +f(y))a m (x"* - l)yf.M(y) + akpx^yftliy)

/«..«\ , ^ 2 2 kp ()'p) j'p /lP>/ \(119) + Cn,ikpx yfa-2 (y) - (y - l)x/„ (x)

. IP Ikp-)

- ajpy xfa-i (x)

.2 2 jp (kp) 3

- Cn,2] P y Xfa-i (X) (mod p ).

Set x = X, a ¿th root of unity, and y = p, a/th root of unity, let p be a prime

greater than a, and then sum each member over all values of f and p except

when f and p are simultaneously 1. We obtain

(120) Zr4(/(-a)+/<-(p))-Zi^^ + ̂ ^} (-d,).

We note that this is symmetric in / and k, hence we need consider only the

first term on the right (call it 7i) and obtain 72, the second term, by inter-

changing/ and k. First, in 7i we sum l/(f —p) with respect to jT for p^ 1. To

effect this we employ the identity

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526 H. S. VANDIVER [May

xk - 1 = (x - l)(x - f) • - • (x - f*-1),

where f now represents a primitive ¿th root of unity. Ordinary differentiation

of this with respect to x gives

x* — 1 x* — 1 xk — 1kxk~l =-1-!-••• +

x— 1 x — f x — f *_1

Dividing by xk — 1; putting x = p; and changing signs, we have

*p*-> 1 1 1-=-+- + ... + -,

1 - p* 1 - p f - P Ï"-1- P

which is the desired sum. Hence

(121) r, = ¿Z' Pfa-l{p) + 12' f"-l() ■, Pd-Pk) T P(ç-D

Now, using (71), we have

qjfe/líyq) = _ a¿(¿ - 1) /^(l)

t P(S - 1) 2 #

(122) 1 ,.5^0»=-ak(k— 1)-

2 >

1=;-a¿(¿ — l)jba-x (mod />)

fora</>. To transform the other term in Pi we employ the relation, with «>0,

xp(- l)n/ni—j = Mx) (mod p),

together with(19)

tiÍP)f \ /n-l(p)/» (p) ■» - «7¿-(mod p2).

1 - p"

In this way we obtain for a>2, modulo p,

y, P%1(P) __ y, (a - l)jPkfa-Áp)

, P(l-Pk)~ , (1- pk)(pp~ 1)

m y, (a - D/(- D'-y-py^p-1)' T " (i - p*)(pp - i)

m y, (a - l)j(- Dtt-2/q-2(p-1)

= , " (P"* - D(l - P"")

(") Vandiver, Annals of Mathematics, (2), vol. 27 (1926), p. 175.

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1942] THE BERNOULLI NUMBERS 527

that is,

(123, £.iV^w , z, (. - w- »-/-.m.id - /.') , (p> -1)(1 - P-)

Using (122) and (123) in connection with (121), we obtain

a¿2(a - l)/(- l)°-2/a-2(p) ak(k -1)Ti m 2J-—-—-jba-i (mod p),

(p* - 1)(1 - pp) 2

and 72 is obtained from this by interchanging/ and k together with p and f.

Add the quantity

(124) — [(/(*"'(!) + fUp)(l)Y = [(b + b'Yjk (mod p)P2

to both members of (120); after employing

Zfnkp\p) = kn+\p(modp2)

f

and the corresponding relation involving/, we then obtain (using (95))

[(kb+jb'Y ba-l ,.,..».+ T.-77 (* + 7) H-

a(a - 1) 2(a - 1) a(125)

t^ (- D°/a-2(p) ,^, (- i)%-2(0 , A ^m k/\J-Y JzJ-(mod p),

r (p* - D(i - pp) t cr* - D(i - fp)where/>1, &>1, 2<a<p, with/, £ and £ prime each to each. Employing

(68) for x = p, n = a — 2, we obtain the following theorem:

Theorem V. If j >l,k>l,a>2, with (j, k) = 1, p a jth root of unity differ-

ent from 1, f a kth root of unity different from 1, then

[(kb+jb'Y ba-l b.+ —-— (* +j) +

a(a — 1) 2(a — 1) a(126)

pRa-2(p) ._ f2e_.(i-)= *2- -z-it,-r^zr +Jls-

(i - p")ip - i)-> T (i - f0(f - i)-1

w&ere the summations extend over each distinct value of p and f, respectively, and

the R's are defined as in (13).

We shall now show how to obtain (91) from (125), but subject to the re-

strictions on the latter relation. Using (71) we have

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528 H. S. VANDIVER [May

and letting c, as before be the least positive or zero residue of ik (mod/), and

employing (65), we have

(a-i)/(-i)-2/„_2(p) ä.r,.., ,aX , j(j-u, .A,(127) 22 ——-—-— = 22*\(jb + Ci)"-1 - ba-x-(mod p),

(pk - 1)(1 - pp) i,x L 2

with a similar expression for the other term in the right-hand member of

(125); if we substitute in (125) we have (91), after noting that the resulting

terms are independent of p.

Using Theorem V, we obtain the corollary:

Corollary I. The expression

([(kb+jb'Y ba-x K\

J V a(a-l) +2(a-l){ +J)+ a)

is an integer with the restrictions onj, k and a given in Theorem V.

We also have this corollary:

Corollary II. If j, k and a are restricted as in Theorem V, we have,

if (jk, a(a-l)) = l,

2[(kb +jb'Y + aba-x(k +j) + 2(a - l)ba = 0 (mod a(a - 1)).

These results indicate certain analogies between the properties of Tn in

(69) and the number expressed by the left-hand member of (125).

16. Bernoulli numbers of higher order. Bernoulli numbers of higher order

than the second, namely, numbers of the form

[(mTbM + mr_iè(r-1) + • • • + «ií' + m0)n = b„(tnr, mT-x, • • ■ , m0),

(128)for r ^ 3; mi 9* 0; i = 1, 2, • • • , r,

do not have properties as simple as those of the first and second order, since

other primes than the von Staudt-Clausen primes appear as factors in the de-

nominators of such numbers. This may be illustrated in the case of the num-

ber

[(/j(l) + ¿(2) ^-_|_ /j(.))».

This may be reduced as follows. Consider first

[(b + b' + b"Y =bn + nbn-x(b' + b") + ■■■ +Cn,2bn-i(b' + b")2 + ■ • • .

Then

[(b> + b")k= (1 - k)bk- kbk-x,

so that

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1942] THE BERNOULLI NUMBERS 529

\(b + V + b"Y = zZCn,rb'n-r((l - r)br - rbr-i)L r-0

n n n

(129) = ¿ZCn.rbn-rbr ~ zZ Cn,rrb'n-rbr ~ zZ C„,rr¿)n_r*)r_l.r=0 r=0 r—0

Now

(130) JZCn.rb'n-rbr=\ib + b'Y.r-0 L

Also,

(131) ¿ rCnX-rbr = « Z C n-l .r-X-A-lb = « £ [*(* + b')"1.r=0 r=l L

Now

[bib + Ô')-1 = (1/2) [ib + b')ib + b')^ = (1/2) [ib + b'Y,

so that

(132) ¿ rCnX-rbr = 4 [(* + *')r-r-0 2 L

Also,

n n p

(133) £C»,rr6^_ri_.i = «E Cn-i.T-ibn-X-i = » (ô + i')*^1.r-0 r=l L

Hence, using (130), (131), (132), and (133) with (129), we have

(134) lib + V + b"Y = (l -—j\ ib + b'Y - nïib + b'Y'1.

We shall now prove, by induction on s, the formula (w>0)

|(ô(i)+ô(2)+ . . . +j(«))» = ( i-—)\(bw+bm+ . . . +&(-i>)»(135) L \ J-l/L

-n[(bw+b<-2)+ ■ ■ ■ +b<-"-l)Y-i.

Assume for 5>2

|(J(D+Ô(2)+ . . . +ô(»-l))"=( 1-,— \\ibW+l,m+ . . . +JÍ.-2))»(136) L \ s — 2/\_

-n[ib^+b™ + • • • +6(-«)—i;then

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530 H. S. VANDIVER [May

(137) [(bw+b^+ ■ • • +¿w)-= ¿c,,r[Vr(ô<i>+ • ■ • +b^-»y,

and by (144)

= T,Cn,r\bn-r(bW+--- +*«-*>)'r-0 L

- Z ^—Cn,r\bn-r(b^+ ■ ■ • + ô<«"2>)'r=o 5 — 2 L

- Z rC„rL(J(«+ • • • +*<-»)-i;r=0 L

or by previous methods this reduces to

[(*' H-+ ¿Cs))n

= \(b'-\-+ $<-»>)«

» r-(¿,(1) + . . . + ¿)(«-2))(6(l) + - • - + /j(«-D)n-l

(138) 5-2L

- Z «Cn_,,T-ir¿.„_T(6' + • • • + ô««-2')»-1r=l L

= \(b' H-+ ô'-1))»-—^- (M" H-+ i«-»)»L Í — 2 5 — 1

- «[(&<» + • • • + 6<-»)n,

which is the result.

Employing (136), we find

(139) [(6 + b' + b"Y = \(n - 1)(» - 2)b„ + !«(« - 2)6„_i + «(» - l)ô„_2.

Repeated application of (135) gives easily

[1 r b -■(b+b'+ ■ ■ ■ +b^)n=(-iy— n(n-l)(n-2) ■ ■ ■ (n-r) Z °n->

rl i_oi—o « — i

where the o>< are the elementary symmetric functions of the numbers

1,2,3, •■• ,r, taken * at a time(20).

(2o) "phis result is due to Lucas, Bulletin de la Société Mathématique de France, vol. 6

(1877), pp. 57-68. The proof given here is due to Dr. A. M. Mood. The proof of relation (144)

was found independently by the writer.

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1942] THE BERNOULLI NUMBERS 531

Suppose n^r in (140); then all the terms will be zero except the one for

i = n, hence

Ub + b' + b"Y= i- l)r-^(- l)*-(r -»)!*«,

hence

(141) Cm = (- l)"Cr.n(b + b' + ■ ■ ■ + &<'>)".

This gives another form for (148) as follows:

[(*+*'+ • • • +b"Y

(142)

= i-lY— «(«-1) • • • (f»-r)¿ (-l)'CrJ(ô+6'+ • • • +*('>)<—■ri i=o L »—»

University of Texas,

Austin, Texas.

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