concrete mathematics a foundation for computer science phần 10 docx

Leonhard Euler, “Methodus generalis summandi progressiones,” Com-mentarii academiz scientiarum Petropolitana 6 1732, 68-97.. Leonhard Euler, “De progressionibus harmonicis observationes

values of X in a sequence of independent trials will be a median or mode ofthe random variable X.

8.53 We can disprove the statement, even in the special case that eachvariableisOor1 LetpO=Pr(X=Y=Z=O),pl=Pr(X=Y=Z=O), ,p7=Pr(X=Y=Z=O),whereX=l-X Thenpo+pl+ +p7=1,andthe variables are independent in pairs if and only if we have

This is equivalent to flipping two fair coins and letting X = (the first coin

is heads), Y = (the second coin is heads), Z = (the coins differ) Anotherexample, with all probabilities nonzero, is

P O = 4/64, PI = ~2 = ~4 = 5/64,

p3 = p5 = p6 = 10/64, p7 = 15/64

For this reason we say that n variables XI, , X, are independent if

Pr(X1 =x1 and and Xn=x,) = Pr(X, =xl) Pr(X, = x , ) ;

pairwise independence isn’t enough to guarantee this

8.54 (See exercise 27 for notation.) We have

E(t:) = nll4 +n(n-1)~:;

E(LzLfI = np4 +2n(n-l)u3pl +n(n-1)~: +n(n-l)(n-2)p2&;E(xy) = np4 +4n(n-l)p~u1 +3n(n-1)~:

+ 6n(n-l)(n-2)u2p: + n(n-l)(n-2)(np3)pT ;

it follows that V(\iX) = K4/n + 2K:/(n ~ 1)

8.55 There are A q = & 52! permutations with X = Y, and B = g 52!permutations with X # Y After the stated procedure, each permutationwith X = Y occurs with probability A/(( 1 - gp)A), because we return

to step Sl with probability $p Similarly, each permutation with X # Yoccurs with probability g(l - p)/((l ~ sp)B) Choosing p = i makes

Pr (X = x and Y = 9) = & for all x and y (We could therefore make two flips

of a fair coin and go back to Sl if both come up heads.)

564 ANSWERS TO EXERCISES

8.56 If m is even, the frisbees always stay an odd distance apart and the

game lasts forever If m = i:l + 1, the relevant generating functions are

(The coefficient [z”] Ak is the probability that the distance between frisbees

is 2k after n throws.) Taking a clue from the similar equations in exercise 49,

we set z = 1 /cos’ 8 and Al := X sin28, where X is to be determined It follows

by induction (not using the equation for Al) that Ak = X sin2kO Therefore

we want to choose X such that

4cos28 > X sin;!10 = 1 + & X sin(21- 218.

It turns out that X = 2 cos’ O/sin 8 cos(21+ 1 )O, hence

c o s e

G, =

-c o s me ’The denominator vanishes when 8 is an odd multiple of n/(2m); thus 1 -qkz is

a root of the denominator for 1 6 k 6 1, and the stated product representation

must hold To find the mean and variance we can write Trigonometry wins

i(m2-1) andVar(G,) = im2(m2-1) (Note that thisimplies theidentities

m2 - 1 (m-1 l/2

~ =

2 & i = 'mjf'2(l/sin "",r,,')")';

k=lm2(m2 - 1) lm Ii/2-

cot (2k- 1)~ (2k- 1)n 2

2 m I sin 2m > ’k=l

The third cumulant of this distribution is &m2(m2 - l)(4m2 - 1); but the

pattern of nice cumulant factorizations stops there There’s a much simpler

way to derive the mean: We have G, + Al + + Ar = z(Ar + + AL) + 1,hence when z = 1 we have Gk = Al + + Al Since G, = 1 when z = 1, aneasy induction shows that Ak = 4k.)

8.57 We have A:A 3 2’ ’ and B:B < 2l ’ + 2l 3 and B:A 3 2’ 2, hence13:B - B:A 3 A:A - A:B is possible only if A:B > 21p3 This means that

52 = ~3, ~1 = ~4, ~2 = ‘~5, , rr 3 = rr But then A:A zz 2’ ’ + 2’ 4 + IA:B z 2’ 3 + 2’ 6 + , B:A z 2’ ’ + 2’ 5 + , and B:B z 2’ -’ + 2’ 4 + ;

hence B:B - B:A is less than A:A - A:B after all (Sharper results havebeen obtained by Guibas and Odlyzko [138], who show that Bill’s chances arealways maximized with one of the two patterns H-r1 rl I or Trl rl , )8.58 According to r(8.82), we want B:B - B:A > A:A - A:B One solution is

m (rn-cwzJk lwZ~m.:;S k-lZ.z

(b) We can either argue algebraically, taking partial derivatives of G (w, z)with respect to w and z and setting w = z = 1; or we can argue com-binatorially: Whatever the values of hl, , h, -1, the expected value ofP(hl , , h, 1, h,; n) is the same (averaged over h,), because the hash se-quence (hr , , h, 1) determines a sequence of list sizes (nl , n2, , n,) suchthat the stated expected value is ((nr+l) + (nz+l) + + (n,+l))/m =(n - 1 + m)/m Therefore the random variable EP( hl , , h,,; n) is indepen-dent of (hl , , h, I), hence independent of P( hr , , h,; k).

8.60 If 1 6 k < 1 :$ n, the previous exercise shows that the coefficient ofsksr in the variance of the average is zero Therefore we need only considerthe coefficient of si, which is

t

Pih,, ,h,;k)2-’

1Sh1 , I h,,Sm mn

l<h, , I h,,<mthe variance of ((m - 1 + z)/m) k~’ z; and this is (k - l)(m - 1)/m’ as inexercise 30

8.61 The pgf D,(z) satisfies the recurrence

Do(z) = z;

b(z) = z2Dn I (2) + 2(1 - z3)Dk -, (z)/(n + 1)) for n > 0

a n d ~4 + 3~: = E ( ( X - ~1~) must be at least (E((X - FL)‘))’ = K:, etc.

A necessary and sufficient condition for this other problem was found byHamburger [6], [144].)

8.63 (Another question asks if there is a simple rule to tell whether H or T

is preferable.) Conway conjectures that no such ties exist, and moreover thatthere is only one cycle in the directed graph on 2’ vertices that has an arcfrom each sequence to its “best beater!’

9.1 True if the functions are all positive But otherwise we might have,say, fl (n) = n3 + n2, fz(n) = -n3, g1 (n) = n4 + n, g2(n) = -n4

9.2 ( a ) W e h a v e nlnn 4 c” 4 (Inn)“, since (lnn)2 + nlnc 4 n l n l n n (b) nlnlnlnn 4 (Inn)! + nlnlnn (c) Take logarithms to show that (n!)! wins.(4 ‘$,,, =: 42lnn = ,2lnl$ HF, ,,-nln$winsbecause@‘=@+l <e.9.3 Replacing kn by 0 (n) requires a different C for each k; but each 0stands for a single C In fact, the context of this 0 requires it to stand for

a set of functions of two variables k and n It would be correct to write,Tc=, kn = EL=, O(n2) = O(n3)

9.4 For example, limn+03 0(1/n) = 0 On the left, 0(1/n) is the set of allfunctions f(n) such that there are constants C and no with If(n)1 < C/n forall n 3 no The limit of all functions in that set is 0, so the left-hand side isthe singleton set {O} On the right, there are no variables; 0 represents {0}, the(singleton) set of all “ functions of no variables, whose value is zero!’ (Can yousee the inherent logic here? If not, come back to it next year; you probablycan still manipulate O-notation even if you can’t shape your intuitions intorigorous formalisms.)

9.5 Let f(n) = n2 and g(n) = 1; then n is in the left set but not in theright, so the statement is fa.lse

9.6 nlnn+yn+O(filnn)

9.7 ( 1 -em’/n)P’ =nBo-B1 +B2n~~‘/2!+~.~=n+~+O(n ‘)

9.8 For example, let f(n) = [n/2]!’ +n, g(n) = ([n/2] - l)! [n/2]! +n.These functions, incidentally, satisfy f(n) = O(ng(n)) and g(n) = O(nf(n));more extreme examples are clearly possible

9.9 (For completeness, we assume that there is a side condition n + 00,

so that two constants are implied by each 0.) Every function on the left hasthe form a(n) + b(n), where there exist constants Q, B, no, C such thatla(n)/ 6 Blf(n)[ for n 3 mc and [b(n)1 6 Clg(n)l for n 3 no Therefore theleft-handfunctionisatmostmax(B,C)(lf(n)l+Ig(n)l),forn3max(~,no),

so it is a member of the right side

9.10 If g(x) belongs to the left, so that g(x) = cosy for some y, whereIy/ < Clxl for some C, then 0 6 1 - g(x) = 2sin2(y/2) < $y2 6 iC2x2; hencethe set on the left is contained in the set on the right, and the formula is true

9.11 The proposition is true For if, say, 1x1 < /yI, we have (x + Y)~ 6 4y2.Thus (x+Y)~ = 0(x2) +O(y’) Thus O(x+y)’ = O((x+y)‘) = 0(0(x2) +O(y2)) = 0(0(x2)) -t O(O(y2)) = 0(x2) + O(y2)

9.12 1 +2/n + O(nP2) = (1 + 2/n)(l + O(nP2)/(1 +2/n)) by (g.26), and

l/(1 +2/n) = O(1); now use (9.26)

9 1 3 n”(1 + 2nP’ + O(nP2))” = nnexp(n(2n-’ + O(nP2))) = e2nn +O(n”-‘)

9.14 It is nn+Pexp((n+ @)(ol/n- ta2/n2 +O(ne3)))

9.17 L>OB,(i)z."'/m! = ~e~'~/(e~-l) = z/(eZ/2-1)-z/(e"-1)

9.18 The text’s derivation for the case OL = 1 generalizes to give

568 ANSWERS TO EXERCISES

9.19 Hlo = 2.928968254 z 2.928968256; lo! =I 3628800 z 3628712.4; B,,., =

0.075757576 z 0.075757494; n( 10) = 4 z 10.0017845; e".' = 1.10517092 z 1.10517083;ln1.1 = 0.0953102 z 0.0953083; 1.1111111 z 1.1111000~ l.l@.' = 1.00957658 z 1.00957643 (The approximation to n(n) gives more significantfigures when n is larger; for example, rc( 1 09) = 50847534 zz 50840742.)

9.20 (a) Yes; the left side is o(n) while the right side is equivalent to O(n).(b) Yes; the left side is e eoi’/ni (c) No; the left side is about J;; times thebound on the right

hi n (lnn)2 + O(l/logn)’ .)(A slightly better approximation replaces this 0( l/logn)’ by the quantity-5/(lnn)’ + O(loglogn/logn)3; then we estimate P~OOOOOO z 15483612.4.)

9.22 Replace O(nzk) by &npLk + O(n 4k) in the expansion of H,r; thisreplaces O(t3(n2)) by -h.E3(n2) + O(E:3(n4)) in (9.53) We have

,X3(n) = ii-i- ’ + &n,F2 + O(np3),

hence the term O(n2) in ($1.54) can be replaced by -gnp2 + O(n 3).g.23 nhn = toskcn hk/(n~-k) +ZcH,/(n+ l)(n+2) Choose c = enL/6 =

tkaogk so that tka0 hk := 0 a n d h, = O(log n)/n3 T h e e x p a n s i o n o f

t OSk<n hk/(n - k) as in (9.60) now yields nh, = ZcH,/(n + l)(n + 2) +O(n m2), hence

which is 2O(f(n) tkzO If(k)/), so this case is proved (b) But in this case if

an -- b, = aPn, the convolution (n + 1 )aPn is not 0( 01 “)

9.25 s,/(3t) = ~;4Lq2n+l)F we may restrict the range of summation

to 0 < k 6 (logn)‘, say In this range nk = nk(l - (i)/n + O(k4/n2)) and(2n + l)k = (2n)k(l + (“;‘)/2n+ O(k4/n2)), so the summand is

Hence the sum over k is 2 -4/n + 0( 1 /n2) Stirling’s approximation can now

be applied to (y) = (3n)!/(2n)!n!, proving (9.2)

9.26 The minimum occurs at a term Blm/(2m) (2m- 1 )n2”-’ where 2m z2rrn + 3, and this term is approximately equal to 1 /(rceZnnfi ) The absoluteerror in Inn! is therefore too large to determine n! exactly by rounding to aninteger, when n is greater than about e2n+‘

9.27 We may assume that a #

where A z 1.282427 is “Glaisher’s constant!’

9.29 Let f(x) = xP1 lnx Then fiZmi (x) > 0 for all large x, and we can write

n Ink

k=l

where S z 0.929772 is constant Taking exponentials gives

(In general if f(x) = X~ lnx, Euler’s summation formula applies as in cise 27, and the resulting constant is -<‘(-a) if a # -1 Thus, the theory ofthe zeta function gives a closed form for Glaisher’s constant in the previousexercise We have 1nS = yi in the notation of answer 9.57.)

Since g(x) = x1 - x2+‘/l ! + x4 ‘l/2! - x6+‘/3! + , the derivatives g imi (x) obey

a simple pattern, and the answer is

9.31 The somewhat surprising identity l/(cmmmk + cm) + l/(~"'+~ + cm) =

1 /cm makes the terms for 0 < k 6 2m sum to (m + +)/cm The remaining

terms are

=-C2m+l _ C2m - C3m+2 _ C3m + )

and this series can be truncated at any desired point, with an error not

ex-ceeding the first omitted term

9 3 2 H:) = x2/6 - l/n + O(nP2) by Euler’s summation formula, since we

know the constant; and H, is given by (9.89) So the answer is The world’s top

three constants,

ney+nL’6 1 - in-’ + O(n-‘)) ( (e, n, y), all appear

in this answer.

9.33 Wehavenk/n’= l-k.(k-l)nP’+~k2(k l)2n~2+0(k6nP3); dividing

by k! and summing over k 3 0 yields e - en-’ $- I en- 2 + 0 ( nP3 )

9.34 A = ey; B = 0; C = -.ie’; D = ieY(l -y); E = :eY; F = &eY(3v+l).

9.35 Since l/k(lnk+ O(l]) = l/kink+ O(l/k(logk)2), t h e g i v e n s u m

is Et==, 1 /kink + 0( 1) The remaining sum is In Inn + 0( 1) by Euler's

summation formula

9.36 This works out beautifully with Euler’s summation formula:

n2 + x2 2 n2 + x2 o +?(n2+x2)2 o + O(nm5)

Hence S, = a7m-l inP2 - An3 + O(nP5).

9.38 Replace k by n - k and let ok(n) = (n - k)nPk(f;) Then In ok(n) =nlnn - Ink! - k + O(kn’), and we can use tail-exchange with bk(n) =nnePk/k!, ck(n) = kbk(n)/n, D, = {k 1 k < lnu}, t o g e t I& o k ( n ) =nne’/e(l + O(n’))

9.39 Tail-exchange with bk(n) = (Inn - k/n - ik2/n2)(lnn)k/k!, ck(n) =n3 (In n) k+3/k!, D, = {k 1 0 < k < 10lnn) W h e n k x 1Olnn w e h a v ek! x fi(lO/e)k(lnn)k, so the kth term is O(n- 101n(lO/e) logn) The answer

i s n l n n - l n n - t(lnn)(l +lnn)/n+O(n~2(logn)3)

9.40 Combining terms two by two, we find that H&-(H2k-&)m = EHykP’plus terms whose sum over all k > 1 is 0 (1) Suppose n is even Euler’ssummation formula implies that

hence the sum is i H,” + 0 (1) In general the answer is 5

(-9.41 Let CX= $/L$ = -@-2 We have

(In eYn)m

+0(l)m

-l)nH,m -t O(1)

ClnFk = ~(h~k-h&+h(l -ak))

k=l

n(n + 1)z

2 In@-5ln5+tln(l -ak)-xln(l -elk).

The latter sum is tIk>,, O(K~) = O(~L~) Hence the answer is

@+1/25-Wc + o&n’” 31/+-n/Z) , where

C = (1 -a)(1 -~~)(l -K~) zz 1.226742

ap-9.43 The denominator has factors of the form z - w, where w is a complexroot of unity Only the factor z - 1 occurs with multiplicity 5 Therefore

by (7.31), only one of the roots has a coefficient n(n4), and the coefficient is

in which each coefficient of xm~k is a polynomial in (x Hence x “x!/(x - CX)! =

Co(R) +c1(a)x ’ + + c,(tx)xpn + 0(x-” ‘) as x + 03, where c,,(a) is apolynomial in 01 We know that c, ( LX) = [,*,I (-1)" whenever 01 is an integer,

and LA1 is a polynomial in 01 of degree 2n; hence c, ( CX) = [ &*,,I (-1)” forall real 01 In other words, the asymptotic formulas

generalize equations (6.13) and (6.11), which hold in the all-integer case.9.45 Let the partial quotients of LX be (a,, al, ), and let cc,,, be the con-tinued fraction l/(a, + CX,,~,) for m 3 1 Then D(cx,n) = D(cxl,n) <D(olr, LarnJ) + al +3 < D(tx3, LcxzlcxlnJj) + al + a2 $6 < < D(Lx,+I,

~~m~ ~~,n~ ~~)+a~+~ +a,+3m<oll cx,n+al+ +a,+3m,

for all m Divide by n and let n + co; the limit is less than 011 CX, forall m Finally we have

A truly Be/l-shaped giving a fairly sharp asymptotic estimate

summand.

-The requested formula follows, with relative error 0 (log log n/log n)

9 4 7 Letlog,n=l+El,whereO$8<1 (ml+’ - l)/(m - 1): the ceiling sum is (L + 1)n - (ml+’ - l)/(m - 1); theexact sum is (1+ 0)n ~ n/in m + O(log n) Ignoring terms that are o(n), thedifference between ceiling and exact is ( 1 - f (0)) n, and the difference betweenexact and floor is f(O)n, where

Thefloorsumisl(n+l)+l-f(e) = J!&Y+e . 1

l n mThis function has m,aximum value f (0) = f (1) = m/( m - 1) - 1 /In m, and itsminimum value is lnlnm/lnm + 1 - (ln(m - l))/ln m The ceiling value iscloser when n is nearly a power of m, but the floor value is closer when 8 liessomewhere between 0 and 1.

9.48 Let dk = ok + bk, where ok counts digits to the left of the decimalpoint Then ok = 1 + Llog Hk] = log log k + 0( 1 ), where ‘log’ denotes loglo

To estimate bk, let us look at the number of decimal places necessary todistinguish y from nearby numbers y e and y + E’: Let 6 = 10 ' be the

574 ANSWERS TO EXERCISIES

length of the interval of numbers that round to 0 We have /y -01 6 id; also

y-e < Q i6 andy+c’ > Q-t-:8 Therefore e+c’ > 6 Andif 6 < min(e, E’),

the rounding does distinguish ij from both y - e and y + 6’ Hence 10Ph” <

l/(k-l)+l/kand 10IPbk 3 l/k; we have bk = log k+O(l) Finally, therefore,

Et=, dk = ,& (logk+loglogk+O(l)), which is nlogn+nloglogn+O(n)

by Euler’s summation formula

9.49 We have H, > lnn+y+ in-’ - &nP2 = f(n), where f(x) is increasing

for all x > 0; hence if n 3 ea Y we have H, 3 f(e”-Y) > K Also H,-, <

Inn + y - in ’ = g(n), where g(x) is increasing for all x > 0; hence if

n 6 eaPy we have H,-l $ g(e” Y) < 01 Therefore H,-r < OL 6 H, implies

t h a t eaPv+l >n>ea+Y-l (Sharper results have been obtained by Boas

and Wrench [27].)

9.50 (a) The expected return is ,YlsksN k/(k’HE’) = HN/H~‘, and we

want the asymptotic value to O(N-’ ):

1nN +y+O(N-‘) 6lnlO 6y 3 6 1 n 1 0 n

n2/6-N-l+O(N-2) = ~n+~~+~~+o(lo-n)*

The coefficient (6 In 1 O)/n2 = 1.3998 says that we expect about 40% profit

(b) The probability o:f profit is x,,<kGN l/(k2Hc’) = 1 - Hf’/HE’,

and since Hf) = $ -n-l + in-’ + O(nm3) this is

n-’ - in2 +O(nP3) 6 -, 3 ~~

nn2/6+ O(N-1) = 7crn + 2+0(nP3),

actually decreasing with n (The expected value in (a) is high because it

includes payoffs so huge that the entire world’s economy would be affected if

they ever had to be made.)

9.51 Strictly speaking, this is false, since the function represented by O(xP2)

might not be integrable (It might be ‘[x E S]/x”, where S is not a measurable

set.) But if we stipulate that f(x) is an integrable function such that f(x) = (As opposed to an

O(xm2) as x + 00, then IJ,“f(x) dx( < j,“lf(x)I dx < j,” CxP2 dx = Cn’ execrable function.)9.52 In fact, the stack of n’s can be replaced by any function f(n) that

approaches infinity, however fast Define the sequence (TQ, ml , ml, ) by

setting rnc = 0 and letting mk be the least integer > mk-1 such that

3 f ( k + 1)‘

Now let A(z) = tk>, (z/k)mk This power series converges for all z, because

the terms for k > Iz/ are bounded by a geometric series Also A(n + 1) 3

((n+ l)/n)“‘n 3 f(n+l)‘, hence lim,,,f(n)/A(n) =O

9.53 By induction, the 0 term is (m - l)! ’ s,” tmP’f(“‘)(x - t) dt Sincef(ln+‘) has the opposite sign to fcm), the absolute value of this integral isbounded by If(“‘(O) 1 J,” tm-’ dt; so the error is bounded by the absolute value

of the first discarded term

9.54 Let g(x) =~f(x)/xrx Then g’(x) N -oLg(x)/x as x t 00 By the meanSounds like a nasty value theorem, g(x - i) - g(x + i) = -g’(y) - ag(y)/y for some y betweentheorem x - i and x + i Now g(y) = g(x)(l +0(1/x)), so g(x - i) - g(x + i) -

ag(x)/x = af(x)/xlta Therefore

x

f(k)

~ =k3n k’+”

(J(t(g&- :I - g(k+ iI)) = o(g(n- :I)

k3n9.55 The estimate of (n + k + i) ln(l + k/n) + (n - k + i) ln(1 - k/n) isextended to k2/n + k4/6n3 + O(nP3/2+5E), so we apparently want to have anextra factor ePk4/6n3 in bk(n), and ck(n) = 22nn-2+5eePk*/n But it turnsout to be better to leave bk(n) untouched and to let

9.57 Using the hini;, the given sum becomes J,” ueCU<( 1 + u/inn) du Thezeta function can be defined by the series

<(l + 2) = C’ + x (-l)“r,z’“/m! ,

Ill>0where yo = y and y,,, is the Stieltjes constant

H e n c e the given sum is

e2niz0 2rriz

Zm+l (Bo + B1 T $- .>

27riz

= &,(Wi +W+ +-.) ,namely (2ti)“‘B,(O)/m! Therefore the sum of residues inside the contour is

m,B,(B) + 2F(27ri)m enim/2 COS (2nk6 nm/2)

+ 1024n3 32768n4 + O(C5)

_-by exercise 5.22 Hence the result follows from exercises 6.64 and 9.44

9.61 The idea is to make cr “almost” rational Let ok = 22zk be the kthpartial quotient of 01, and let n = ;a,,,+, qm, where qm = K(al, , a,) and

m is even Then 0 < {q,,,K} < l/Q(al, ,a,+.,) < 1/(2n), and if we take

v = a,,,+1 /(4n) we get a discrepancy 3 :a,+, If this were less than n’-’ wewould have

E

%+1 = WlAy),

but in fact a,+1 > 42,"

“The paradox

is now fully

es-tablished that

the utmost

abstractions are the

true weapons with

n+‘u(lnlnn/ln 4)

Innsatisfies this recurrence asymptotically, if u(x + 1) = -u(x) (Vardi conjec-tures that

f ( n ) = nml(c+u(c)(lnn)-’ +O((logni’))

for some such function u.) Calculations for small n show that f(n) equals thenearest integer to cn.+’ for 1 6 n < 400 except in one case: f(273) = 39 >

c.273‘+' zz 38.4997 But the small errors are eventually magnified, because

of results like those in exercise 2.36 For example, e(201636503) M 35.73;

e(919986484788) z 1959.07.

9.64 (From this identity for Bz(x) we can easily derive the identity of

exer-cise 58 by induction on m.) If 0 < x < 1, the integral si” sin Nti dt/sin ti

can be expressed as a sum of N integrals that are each 0 (N 2), so it is 0 (N -’ );the constant implied by this 0 may depend on x Integrating the identity

~:,N=lcos2n7rt=!.R(e2"it(e2N"it-l)/(e 2Rit-l)) = -i+i sin(2N+l)ti/sinrrt

and letting N + 00 now gives xnB1 (sin 2nrrx)/n = 5 - XX, a relation thatEuler knew ([85’] and [88, part 2, $921) Integrating again yields the desiredformula (This solution was suggested by E M E Wermuth; Euler’s originalderivation did not meet modern standards of rigor.)

9.65 The expected number of distinct elements in the sequence 1, f(l),

f(f(l)), *, when f is a random mapping of {1,2, , n} into itself, is the

function Q(n) of exercise 56, whose value is i &+O (1); this might account

somehow for the factor v%%

9.66 It is known that lnx,, N in2 In 4; the constant een/6 has been verifiedempirically to eight significant digits

9.67 This would fail if, for example, en-y = m+ t + e/m for some integer mand some 0 < E < f; but no counterexamples are known

Bibliography

HERE ARE THE WORKS cited in this book Numbers in the margin specify

References to published problems are generally made to the places where much-needed gapsolutions can be found, instead of to the original problem statements, unless in the literature.”

N H Abel, letter to B Holmboe (1823), in his CEuvres CompI&es, first

edition, 1839, volume 2, 264-265 Reprinted in the second edition, 1881,

volume 2, 254-255

Milton Abramowitz and Irene A Stegun, editors, Handbook of

Math-ematical Functions U:nited States Government Printing Office, 1964

Reprinted by Dover, 1965

William W Adams and J L Davison, “A remarkable class of continued

fractions,” Proceedings of the American Mathematical Society 65 (1977),

194-198

A V Aho and N J A Sloane, “Some doubly exponential sequences,”

Fibonacci Quarterly 11 (1973), 429-437

W Ahrens, Mathematische Unterhaltungen und Spiele Teubner,

Leip-zig, 1901 Second edition, in two volumes, 1910 and 1918

Naum Il’ich Akhiezer, I<lassicheskal^a Problema Momentov i Nekotorye

Voprosy Analiza, SvI2zannye s Nem Moscow, 1961 English translation,

The classical Moment P.roblem and Some Related Questions in Analysis,

Hafner, 1965

R E Allardice and A Y’ Fraser, “La Tour d’Hanoi’,” Proceedings of the

Edinburgh Mathematical Society 2 (1884), 50-53.

Desire Andre, ‘Sur les permutations alternees,” Journal de

MathCma-tiques pures et appliquCes, series 3, 7 (1881), 167-184

M.D Atkinson, “The cyclic towers of Hanoi,” Information ProcessingLetters 13 (1981), 118-119.

Paul Bachmann, Die analytische Zahlentheorie Teubner, Leipzig, 1894

W N Bailey, Generalized Hypergeometric Series Cambridge UniversityPress, 1935; second edition, 1964

W W Rouse Ejall and H S M Coxeter, Mathematical Recreations andEssays, twelfth edition University of Toronto Press, 1974 (A revi-sion of Ball’s Mathematical Recreations and Problems, first published

by Macmillan, 1892.)

P Barlow, “Demonstration of a curious numerical proposition,” Journal

of Natural Phi,!osophy, Chemistry, and the Arts 27 (1810), 193-205

Samuel Beatty, “Problem 3177,” American Mathematical Monthly 34(1927), 159-160

E T Bell, “Euler algebra,” Transactions of the American MathematicalSociety 25 (1923), 135-154

E T Bell, “Exponential numbers,” American Mathematical Monthly 41(1934), 411-41’9

Edward A Be:nder, “Asymptotic methods in enumeration,” SIAM view 16 (197411, 485-515

Jacobi Bernoulli, Ars Conjectandi, opus posthumum Base& 1713 printed in Die Werke von Jakob Bernoulli, volume 3, 107-286

Re-J Bertrand, ‘%Iemoire sur le nombre de valeurs que peut prendre unefonction quand on y permute les lettres qu’elle renferme,” Journal deI’&ole Royale Polytechnique 18, cahier 30 (1845), 123-140

William H Beyer, editor, CRC Standard Mathematical Tables, 25th tion CRC Press, Boca Raton, Florida, 1978

J Bienayme, “Considerations a l’appui de la decouverte de Laplace sur

la loi de probabilite da.ns la methode des moindres car&,” Comptes

Rendus hebdomadaires des seances de 1’AcadCmie des Sciences (Paris)

37 (1853), 309-324

J Binet, “Memoire sur l’integration des equations lineaires aux

diffe-rences finies, d’un ordre quelconque, a coefficients variables,” Comptes

Rendus hebdomadaires des seances de 1’Academie des Sciences (Paris)

17 (1843), 559-567

Gunnar Blom, “Problem E 3043: Random walk until no shoes,” American

Mathematical Monthly 94 (1987), 78-79

R P Boas, Jr and J W Wrench, Jr., “Partial sums of the harmonic

series,” American Mathematical Monthly 78 (1971), 864-870

P Bohl, “Uber ein in der Theorie der s;ikularen Storungen

vorkom-mendes Problem,” Journal fiir die reine und angewandte Mathematik

135 (1909), 189-283

P du Bois-Reymond, “Sur la grandeur relative des infinis des fonctions,”

Annali di Matematica pura ed applicata, series 2, 4 (1871), 338-353

Bmile Borel, LeGons sur les skies a termes positifs Gauthier-Villars,

1902

Jonathan M Borwein and Peter B Borwein, Pi and the AGM Wiley,

1987

Richard P Brent, “The first occurrence of large gaps between successive

primes,” Mathematics of Computation 27 (1973), 959-963

Richard P Brent, “Computation of the regular continued fraction for

Euler’s constant,” Mathematics of Computation 31 (1977), 771-777

John Brillhart , “Some miscellaneous factorizations,” Mathematics of

Computation 17 (1963), 447-450

Achille Brocot, “Calcul des rouages par approximation, nouvelle

me-thode,” Revue Chronometrique 6 (1860), 186-194 (He also published

a 97-page monograph with the same title in 1862.)

Maxey Brooke and C R Wall, “Problem B-14: A little surprise,”

Cited in A K Austin, “Modern research in mathematics,” The

Mathe-matical Gazette 51 (1967), 149-150

Thomas C Brown, “Problem E 2619: Squares in a recursive sequence,”

American Mathematical Monthly 85 (1978), 52-53.

William G Brown, “Historical note on a recurrent combinatorial lem,” American Mathematical Monthly 72 (1965), 973-977.

prob-S A Burr, “On moduli for which the Fibonacci sequence contains acomplete system of residues,” Fibonacci Quarterly 9 (1971), 497-504

E Rodney Canfield, “On the location of the maximum Stirling ber(s) of the second kind,” Studies in Applied Mathematics 59 (1978)‘83-93

num-Lewis Carroll ‘ipseudonym of C L Dodgson], Through the Looking Glass

and What Alice Found There Macmillan, 1871

Jean-Dominique Cassini, “Une nouvelle progression de nombres,”

His-toire de 1 ‘Acadkmie Royale des Sciences, Paris, volume 1, 201 (Cassini’swork is summarized here as one of the mathematical results presented

to the academy in 1680 This volume was published in 1733.)

E Catalan, “Note sur une aquation aux differences finies,” Journal de

Mathe’matiques pures et appliquCes 3 (1838), 508-516

Augustin-Louis Cauchy, Cours d’analyse de I’ficole RoyaIe

Polytech-nique Imprimerie Royale, Paris, 1821 Reprinted in his CEuvres

Com-PI&es, series 2, volume 3

Arnold Buffum Chace, The Rhind Mathematical Papyrus, volume 1.Mathematical Association of America, 1927 (Includes an excellent bib-liography of Egyptian mathematics by R C Archibald.)

M Chaimovich, G Freiman, a n d J SchGnheim, “ O n e x c e p t i o n s t oSzegedy’s theorem,” Acta Arithmetica 49 (1987), 107-112

P L Tchebichef [Chebyshev], “Mkmoire sur les nombres premiers,” nal de Mathtimatiques pures et applique’es 17 (1852), 366-390 Reprinted

Jour-in his muvres, volume 1, 51-70

P L Chebyshev, “0 srednikh velichinakh,” Matematicheskii Sbornik’ 2(1867), l-9 Reprinted in his Polnoe Sobranie Sochinenii, volume 2, 431-

437 French translation, “Des valeurs moyennes,” Journal de

MathCma-tiques pures et appliqubes, series 2, 12 (1867), 177-184; reprinted in his(Euvres, volume 1, 685-694

Th Clausen, “Beitrag zur Theorie der Reihen,” Journal fiir die reine

und angewandte Mathematik 3 (X328), 92-95

Th Clausen, “Theorem,” Astronomische Nachrichten 17 (1840),

col-umns 351-352

Stuart Dodgson Collingwood, The Lewis Carroll Picture Book T Fisher

Unwin, 1899 Reprinted by Dover, 1961, with the new title Diversions

and Digressions of Lewis Carroll

J H Conway and R.L Graham, “Problem E2567: A periodic

recur-rence,” American Mathematical Monthly 84 (1977), 570-571

Harald Cram&, “On the order of magnitude of the difference between

consecutive prime numbers,” Acta Arithmetica 2 (1937), 23-46

A L Crelle, “DCmonstration ClCmentaire du thCor&me de Wilson

gCnC-ralisC,” Journal fiir die reine und angewandte Mathematik 20 (1840),

29-56

D W Crowe, “The n-dimensional cube and the Tower of Hanoi,”

Amer-ican Mathematical Monthly 63 (1956), 29-30

D R Curt&, “On Kellogg’s Diophantine problem,” American

Mathe-matical Monthly 29 (1922), 380-387

F N David and D E Barton, Combinatorial Chance Hafner, 1962

J L Davison, “A series and its associated continued fraction,”

Proceed-ings of the American Mathematical Society 63 (1977), 29-32

N G de Bruijn, Asymptotic Methods in Analysis North-Holland, 1958;

third edition, 1970 Reprinted by Dover, 1981

N G de Bruijn, “Problem 9,” Nieuw Archief voor Wiskunde, series 3,

500, 510, 603, 604, 7 7

605.

Leonard Eugene Dickson, History of the Theory of Numbers CarnegieInstitution of Washington, volume 1, 1919; volume 2, 1920; volume 3,

1923 Reprinted by Stechert, 1934, and by Chelsea, 1952, 1971

Edsger W Dijkstra, Selected Writings on Computing: A Personal

Per-spective Springer-Verlag, 1982

G Lejeune Dirichlet, “Verallgemeinerung eines Satzes aus der Lehrevon den Kettenbriichen nebst einigen Anwendungen auf die Theorieder Zahlen,” Bericht iiber die Verhandlungen der Koniglich-PreufljschenAkademie der Wjssenschaften zu Berlin (1842), 93-95 Reprinted in hisWerke, volume 1, 635-638

A C Dixon, “On the sum of the cubes of the coefficients in a certain pansion by the binomial theorem,” Messenger of Mathematics 20 (1891),79-80

ex-John Dougall, “On Vandermonde’s theorem, and some more generalexpansions,” Proceedings of the Edinburgh Mathematical Society 25(1907), 114-132

A Conan Doyle, “The sign of the four; or, The problem of the Sholtos,”Lippincott’s Monthly Magazme (Philadelphia) 45 (1890), 147-223

A Conan Doyle, “The adventure of the final problem,” The Strand azine 6 (1893), 558-570

Mag-Henry Ernest Dudeney, The Canterbury Puzzles and Other CuriousProblems E P Dutton, New York, 1908; 4th edition, Dover, 1958 (Du-deney had first considered the generalized Tower of Hanoi in The WeeklyDispatch, on 25 May 1902 and 15 March 1903.)

G Waldo Durmington, Carl Friedrich Gauss: Titan of Science tion Press, New York, 1955

Exposi-A W F Edwards, Pascal’s Arithmetical ‘Triangle Oxford UniversityPress, 1987

G Eisenstein, “Entwicklung von o(“~ ,” Journal fiir die reine und

ange-wandte Mathematik 28 (1844), 49-52 Reprinted in his MathematischeWerke 1, 122-125

Erdos Pal, “AZ k + i + + x, = z e g y e n l e t egesz szamti meg-1oldasairol,” Matematikai Lapok 1 (1950), 192-209 English abstract onpage 210

P Erdijs and R.L Graham, Old and New Problems and Results in

Combinatorial Number Theory Universite de Geneve, L’EnseignementMathematique, 1980

P Erdbs, R L Graham, I Z Ruzsa, and E.G Straus, “On the prime

factors of (:) ,‘I Mathematics of Computation 29 (1975), 83-92

Arulappah Eswarathasan and Eugene Levine, “p-integral harmonic

sums,” Discrete Mathematics, to appear

Euclid, CTOIXEIA Ancient manuscript first printed in Basel, 1533

Scholarly edition (Greek and Latin) by J L Heiberg in five volumes,

Teubner, Leipzig, 1883-1888

Leonhard Euler, letter to Christian Goldbach (13 October 1729), in

Cor-respondance Mathkmatique et Physique de Quelques C&bres GCom&res

du XVIII6me SiWe, edited by P H Fuss, St Petersburg, 1843, volume 1,

3-7

Leonhard Euler, “Methodus generalis summandi progressiones,”

Com-mentarii academiz scientiarum Petropolitana 6 (1732), 68-97

Re-printed in his Opera Omnia, series 1, volume 14, 42-72

Leonhard Euler, “De progressionibus harmonicis observationes,”

Com-mentarii academia? scientiarum Petropolitanae 7 (1734), 150-161

Re-printed in his Opera Omnia, series 1, volume 14, 87-100

Leonhard Euler, “De fractionibus continuis, Dissertatio,” Commentarii

academia: scientiarum Petropolitana: 9 (1737), 98-137 Reprinted in his

Opera Omnia, series 1, volume 14, 187-215

Leonhard Euler, “Varia? observationes circa series infinitas,”

Commen-tarii academiz scientiarum Petropolitana: 9 (1737), 160-188 Reprinted

in his Opera Omnia, series 1, volume 14, 216-244

Leonhard Euler, letter to Christian Goldbach (4 July 1744), in

Corre-spondance Mathkmatique et Physique de Quelques C&bres GCom&res

du XVIIlhme Si&cle, edited by P H Fuss, St Petersburg, 1843, volume 1,

278-293

Leonhard Euler, Introductio in Analysin Infinitorum Tomus primus,

Lausanne, 1748 Reprinted in his Opera Omnia, series 1, volume 8

Trans-lated into French, 1786; German, 1788

Leonhard Euler, “De pactitione numerorum,” Novi commentarii

academ-ia? scientiarum Petropolitana: 3 (1750), 125-169 Reprinted in his

Com-mentationes arithmetic= collect=, volume 1, 73-101 Reprinted in his

Opera Omnia, series 1, volume 2, 254-294

Leonhard Euler, Institutiones Calculi Differentialis cum eius usu in

An-alysi Finitorum ac Doctrina Serierum Petrograd, Academiae Imperialis

Scientiarum, 1755 Reprinted in his Opera Omnia, series 1, volume 10

Translated into German, 1790

Leonhard Euler, “Specimen algorithmi singularis,” Novi commentariiacademia? scientiarum Petropolitanae 9 (1762), 53-69 (Also presented

in 1757 to the Berlin Academy.) Reprinted in his Opera Omnia, series 1,volume 15, 31-49

Leonhard Euler, “Observationes analyticae,” Novi commentarii academirescientiarum Pet.ropolitanae 11 (1765), 124-143 Reprinted in his OperaOmnia, series 1, volume 15, 50-69

Leonhard Euler; Vollsttidige Anleitung BUT Algebra Erster Theil Vonden verschiedenen Rechnungs-Arten, Verhsltnissen und Proportionen

St Petersburg, :1770 Reprinted in his Opera Omnia, series 1, volume 1.Translated into Russian, 1768; Dutch, 1773; French, 1774; Latin, 1790;English, 1797

Leonhard Euler, “Observationes circa bina biquadrata quorum summam

in duo alia biquadrata resolvere liceat,” Novi commentarii academia: entiarum Petropolitana 17 (1772), 64-69 Reprinted in his Opera Om-nia, series 1, volume 3, 211-217

sci-Leonhard Euler, “Observationes circa novum et singulare sionum genus,” Novi commentarii academia scientiarum Petropolitanze

progres-20 (1775), 123-:139 Reprinted in his Opera Omnia, series 1, volume 7,246-261

Leonhard Euler, “Specimen transformationis singularis serierum,” Novaacta academia scientiarum Petropolitana: 12 (1794), 58-70 Submittedfor publication in 1778 Reprinted in his Opera Omnia, series 1, vol-ume 16(2), 41-55

William Feller, An Introduction to Probability Theory and Its tions, volume 1 Wiley, 1950; second edition, 1957; third edition, 1968.Pierre de Ferm,at, letter to Marin Mersenne (25 December 1640), inCIuvres de Fermat, volume 2, 212-217

Applica-Leonardo Fibonacci [Pisano], Liber Abaci First edition, 1202 (now lost);second edition 1228 Reprinted in Scritti di Leonardo Pisano, edited byBaldassarre Boncompagni, 1857, volume 1

Michael E Fisher, “Statistical mechanics of dimers on a plane lattice,”Physical Review 124 (1961), 1664-1672

586 BIBLIOGRAPHY

100 R A Fisher, “Moments and product moments of sampling distribu- 605

tions,” Proceedings of the London Mathematical Society, series 2, 30

(1929), 199-238

101 Pierre Forcadel, L’arithmeticque Paris, 1557 603.

102 J Fourier, “Refroidissement sCculaire du globe terrestre,” Bulletin des 22

Sciences par la Socie’tC philomathique de Paris, series 3, 7 (1820), 58-70

Reprinted in GYuvres de Fourier, volume 2, 271-288

103 Aviezri S Fraenkel, “Complementing and exactly covering sequences,” 500, 602.

Journal of Combinatorial Theory, series A, 14 (1973), 8-20

104 Aviezri S Fraenkel, “How to beat your Wythoff games’ opponent on 538.

three fronts,” American Mathematical Monthly 89 (1982), 353-361

105 J S Frame, B M Stewart, and Otto Dunkel, “Partial solution to prob- 602.

lem 3918,” American Mathematical Monthly 48 (1941), 216-219

106 Piero della Francesca, Libellus de quinque corporibus regularibus Vat- 604.

ican Library, manuscript Urbinas 632 Translated into Italian by Luca

Pacioli, as part 3 of Pacioli’s Diuine Proportione, Venice, 1509

107 W D Frazer and A C McKellar, “Samplesort: A sampling approach to 603.

minimal storage tree sorting,” Journal of the ACM 27 (1970), 496-507

108 Michael Lawrence Fredman, Growth Properties of a Class of Recursively 499.Defined Functions Ph.D thesis, Stanford University, Computer Science

Department, 1972

109 Nikolaus I Fuss, “Solutio quEstionis, quot modis polygonum n lat- 347

erum in polygona m laterum, per diagonales resolvi quzat,” Nova acta

academia: scientiarum Petropolitana: 9 (1791), 243-251

110 Martin Gardner, “About phi, an irrational number that has some re- 285.

markable geometrical expressions,” Scientific American 201,2 (August

1959), 128-134 Reprinted with additions in his book The 2nd Scientific

American Book of Mathematical Puzzles & Diversions, 1961, 89-103.

111 Martin Gardner, “On the paradoxical situations that arise from nontran- 396.sitive relations,” Scientific American 231,4 (October 1974), 120-124 Re-

printed with additions in his book Time Travel and Other Mathematical

Bewilderments, 1988, 55-69

112 Martin Gardner, “From rubber ropes to rolling cubes, a miscellany of 603.

refreshing problems,” Scientific American 232,3 (March 1975), 112-114;

232,4 (April 1975), 130, 133 Reprinted with additions in his book Time

Travel and Other Mathematical Bewilderments, 1988, 111-124

113 Martin Gardner, “On checker jumping, the amazon game, weird dice, 605.

card tricks and other playful pastimes,” Scientific American 238,2

(February 1978), 19, 22, 24, 25, 30, 32

605 114 J Garfunkel, “F)roblem E 1816: An inequality related to Stirling’s

for-mula,” American Mathematical Monthly 74 (1967), 202

123, 602 1 1 5 C F Gauss, Disquisitiones Arithmetic= Leipzig, 1801 Reprinted in his

hypergeo-of the United States hypergeo-of America 75 (1978), 40-42

125 R L Graham, “On a theorem of Uspensky,” American MathematicalMonthly 70 (1963), 407-409

126 R L Graham, “,4 Fibonacci-like sequence of composite numbers,” ematics Magazine 37 (1964), 322-324

Math-127 R L Graham, “Problem 5749,” American Mathematical Monthly 77(1970), 775

588 BIBLIOGRAPHY

128 Ronald L Graham, “Covering the positive integers by disjoint sets of 5~x1.

theform{[nol+fi]:n=1,2, },” Journal of Combinatorial Theory,

series A, 15 (1973), 354-358

129 R L Graham, “Problem 1242: Bijection between integers and compos- 602.

ites,” Mathematics Magazine 60 (1987), 180

130 R L Graham and D E Knuth, “Problem E 2982: A double infinite sum 602.

for 1x1,” American Mathematical Monthly 96 (1989), 525-526

131 Ronald L Graham, Donald E Knuth, and Oren Patashnik, Concrete 102.Mathematics: A Foundation for Computer Science Addison-Wesley,

1989 (The first printing had a different Iversonian notation.)

132 R L Graham and H 0 Pollak, “Note on a nonlinear recurrence related 602

to fi,I’ Mathematics Magazine 43 (1970), 143-145

133 Guido Grandi, letter to Leibniz (July 1713), in Leibnizens mathematische 58.

Schriften, volume 4, 215-217

134 Daniel H Greene and Donald E Knuth, Mathematics for the Analysis 520, 605

of Algorithms Birkhguser, Boston, 1981; third edition, 1990

135 Samuel L Greitzer, International Mathematical Olympiads, 1959-1977 602.

Mathematical Association of America, 1978

136 Oliver A Gross, “Preferential arrangements,” American Mathematical 604.

Monthly 69 (1962), 4-i3.

137 Branko Griinbaum, “Venn diagrams and independent families of sets,” 484.

Mathematics Magazine 48 (1975), 12-23

138 L J Guibas and A M Odlyzko, “String overlaps, pattern matching, and 565, 605

nontransitive games,” Journal of Combinatorial Theory, series A, 30

(1981), 183-208

139 Richard K Guy, Unsolved Problems in Number Theory Springer- 510.

Verlag, 1981

140 Marshall Hall, Jr., The Theory of Groups Macmillan, 1959 530.

141 P.R Halmos, “How to write mathematics,” L’Enseignement matht?ma- vi.tique 16 (1970), 123-152 Reprinted in How to Write Mathematics,

American Mathematical Society, 1973, 19-48

142 Paul R Halmos, I Want to Be a Mathematician: An Automathography v.Springer-Verlag, 1985 Reprinted by Mathematical Association of Amer-

ica, 1988

143 G H Halphen, “Sur des, suites de fractions analogues B la suite de Farey,” 291.

Bulletin de la SociCtC mathkmatique de France 5 (1876), 170-175

Re-printed in his C%vres, volume 2, 102-107

‘Mod-146 J M Hammersley, “An undergraduate exercise in manipulation,” TheMathematical Scientist 14 (1989), l-23.

147 Eldon R Hansen, A Table of Series and Products Prentice-Hall, 1975

148 G H Hardy, Orders of Infinity: The ‘Infinitticalciil’ of Paul du Reymond Cambridge University Press, 1910; second edition, 1924

Bois-149 G H Hardy, “A mathematical theorem about golf,” The MathematicalGazette 29 (1944), 226-227 Reprinted in his Collected Papers, volume 7,488

150 G H Hardy and E M Wright, An Introduction to the Theory of bers Clarendon Press, Oxford, 1938; fifth edition, 1979

Num-151 Peter Henrici, Applied and Computational Complex Analysis Wiley,volume 1, 1974; volume 2, 1977; volume 3, 1986

152 Peter Henrici, “De Branges’ proof of the Bieberbach conjecture: A viewfrom computational analysis,” Sitzungsberichte der Berliner Mathema-tischen Gesellschti (1987), 105-121

153 Charles Hermite, letter to C W Borchardt (8 September 1875), in nal fiir die reine und angewandte Mathematik 81 (1876), 93-95 Re-printed in his QZuvres, volume 3, 211-214

Jour-154 Charles Hermite, Cours de M Hermite FacultC des Sciences de Paris,

1882 Third edition, 1887; fourth edition, 1891

155 Charles Hermite, letter to S Pincherle (10 May 1900), in Annali diMatematica pura ed applicata, series 3, 5 (1901), 57-60 Reprinted inhis CGvres, volume 4, 529-531

156 I.N Herstein and I Kaplansky, Matters Mathematical Harper & Row,1974

157 A P Hillman and V E Hoggatt, Jr., “A proof of Gould’s Pascal hexagonconjecture,” Fibonacci Quarterly 10 (1972), 565-568, 598.

158 C A R Hoare, “Quicksort,” The Computer Journal 5 (1962), 10-15

159 L C Hsu, “Note on a combinatorial algebraic identity and its tion,” Fib0nacc.i Quarterly 11 (1973), 480-484.

applica-590 BIBLIOGRAPHY

160 K Inkeri, “Absch;itzu:ngen fiir eventuelle Lijsungen der Gleichung im 509.Fermatschen Problem,” Annales Universitatis Turkuensis, series A, 16,

1 (1953), 3-9

161 Kenneth E Iverson, A Programming Language Wiley, 1962 24, 67, 602.

162 C G J Jacobi, Fundamenta nova theor& functionurn ellipticarum 64.K6nigsberg, BorntrBger, 1829 Reprinted in his Gesammelte Werke, vol-

ume 1, 49-239

163 Dov Jarden and Theodor Motzkin, “The product of sequences with a 533.common linear recursion formula of order 2,” Riveon Lematematika 3

(1949), 25-27, 38 (Heb’rew with English summary) English version

re-printed in Dov Jarden., Recurring Sequences, Jerusalem, 1958, 42-45;

second edition, Jerusalem, 1966, 30-33

164 Arne Jonassen and Donald E Knuth, “A trivial algorithm whose analysis 520.isn’t,” Journal of Computer and System Sciences 16 (1978), 301-322.

165 Bush Jones, “Note on internal merging,” Software -Practice and Expe- 175.rience 2 (1972), 241-243

166 Flavius Josephus, ETOPIA IOTAAIKOT ITOAEMOT ITPOC Pa- 8.MAIOTC English translation, History of the Jewish War against the

Remans, by H St J Thackeray, in the Loeb Classical Library edition

of Josephus’s works, volumes 2 and 3, Heinemann, London, 1927-1928

(The “Josephus problem” may be based on an early manuscript now

pre-served only in the Slavonic version; see volume 2, page xi, and volume 3,

169 Murray S Klamkin, Initernational Mathematical Olympiads, 1978-1985, 602, 603.

and Forty Supplementary Problems Mathematical Association of

Amer-ica, 1986

170 Konrad Knopp, Theorie und Anwendung der unendlichen Reihen Julius 605.Springer, Berlin, 1922; second edition, 1924 Reprinted by Dover, 1945

Fourth edition, 1947; fifth edition, 1964 English translation, Theory and

Application of Infinite Series, 1928; second edition, 1951

171 Donald E Knuth, “Euler’s constant to 1271 places,” Mathematics of 467.Computation 16 (1962), 275-281

172 Donald Knuth, “Transcendental numbers based on the Fibonacci se- 531.quence,” Fibonacci Quarterly 2 (1964), 43-44, 52.

173 Donald E Knuth, The Art of Computer Programming, volume 1:

Fun-damental Algor.ithms Addison-Wesley, 1968; second edition, 1973

174 Donald E Knuth, The Art of Computer Programming, volume 2:Seminumerical Algorithms Addison-Wesley, 1969; second edition, 1981

175 Donald E Knuth, The Art of Computer Programming, volume 3: Sortingand Searching Addison-Wesley, 1973; second printing, 1975

176 Donald E Knuth, “Problem E 2492: Some sum,” American MathematicalMonthly 82 (1975), 855

177 Donald E Knuth, Mariages s t a b l e s e t leurs relations avec d’autresprobl&mes combinatoires Les Presses de 1’UniversitC de Mont&al, 1976.Revised and corrected edition, 1980

178 Donald E Knuth, The wbook Addison-Wesley, 1984 Reprinted asvolume A of Computers & Typesetting, 1986

179 Donald E Knuth, “An analysis of optimum caching,” Journal of rithms 6 (1985), 181-199

Algo-180 Donald E Knuth, Computers & Typesetting, volume D: METRFONT:

The Program Addison-Wesley, 1986

181 Donald E Knuth, “Problem 1280,” Mathematics Magazine 60 (1987),329

182 Donald E Knuth, “Problem E 3106: A new sum for n’,” American ematical Monthly 94 (1987), 795-797

Math-183 Donald E Knuth, “Fibonacci multiplication,” Applied Mathematics ters 1 (1988), 57-60

Let-184 Donald E Knuth, “A Fibonacci-like sequence of composite numbers,”Mathematics Magazine 63 (1990), 21-25

185 Donald E Knufh and Thomas J Buckholtz, “Computation of Tangent,Euler, and Bernoulli numbers,” Mathematics of Computation 21 (1967),663-688

186 C Kramp, lhmens d’arithmgtique universelle Cologne, 1808.

187 E E Kummer, “Ueber die hypergeometrische Reihe

592 BIBLIOGRAPHY

188 E.E Kummer, “Uber die Erganzungssatze zu den allgemeinen Re- 603.ciprocitatsgesetzen,” Journal fiir die reine und angewandte Mathematik

44 (1852), 93-146 Reprinted in his Collected Papers, volume 1, 485-538

189 R P Kurshan and B Gopinath, “Recursively generated periodic se- 487.

quences,” Canadian Journal of Mathematics 26 (1974), 1356-1371.

190 Thomas Fantet de Lagny, Analyse g&&ale ou Methodes nouvelles pour 290.resoudre les problemes de tous les genres et de tous les degr6 2 l’infini.

Published as volume 11 of Memoires de 1’AcadCmie Royale des Sciences,

Paris, 1733

191 J.-L de la Grange [Lagrange], “Demonstration d’un theoreme nouveau 604.

concernant les nombreis premiers,” Nouveaux Memoires de 1’AcadCmie

royale des Sciences et Belles-Lettres de Berlin (1771), 125-137 Reprinted

in his auvres, volume 3, 425-438

192 J.-L de la Grange [Lagrange], “Sur une nouvelle espece de calcul relatif 456

a la differentiation & a l’integration des quantites variables,” Nouveaux

Memoires de 1’AcadCmie royale des Sciences et Belles-Lettres de Berlin

(1772), 185-221 Reprinted in his Ckvres, volume 3, 441-476.

193 I Lah, “Eine neue Art von Zahlen, ihre Eigenschaften und Anwendung 603.

in der mathematischen Statistik,” Mitteilungsblatt fiir Mathematische

Statistik 7 (1955), 2033212

194 Edmund Landau, Handbuch der Lehre von der Verteilung der Prim- 434, 605.

zahlen, two volumes Teubner, Leipzig, 1909

195 Edmund Landau, Vorlesungen iiber Zahlentheorie, three volumes Hirzel, 603.

Leipzig, 1927

195’ P S de la Place [Laplace], “Memoire sur les approximations des Formules 452.

qui sont fonctions de t&-grands nombres,” Memoires de 1’Academie

royale des Sciences de Paris (1782), l-88 Reprinted in his CEuvres

Completes 10, 207-291

196 Adrien-Marie Legendre, Essai sur la The’orie des Nombres Paris, 1798; 602.

second edition, 1808 Third edition (retitled The’orie des Nombres, in two

volumes), 1830; fourth edition, Blanchard, 1955

197 D H Lehmer, “Tests for primality by the converse of Fermat’s theorem,” 602.

Bulletin of the American Mathematical Society, series 2, 33 (1927),

203 Calvin T Long and Verner E Hoggatt, Jr., “Sets of binomial coefficientswith equal products,” Fibonacci Quarterly 12 (1974), 71-79.

204 Sam Loyd, Cyclopedia of Puzzles Franklin Bigelow Corporation, ingside Press, New York, 1914

Morn-205 E Lucas, “Sur les rapports qui existent entre la theorie des nombres

et le Calcul integral,” Comptes Rendus hebdomadaires des seances deI’AcadCmie des Sciences (Paris) 82 (1876), 1303-1305

206 Edouard Lucas, “Sur les congruences des nombres euleriens et des ficients differentiels des fonctions trigonometriques, suivant un modulepremier,” Bulletin de la SociCtC mathematique de France 6 (1878), 49-54

coef-207 Edouard Lucas, ThCorie des Nombres, volume 1 Gauthier-Villars, Paris,1891

208 Edouard Lucas, Recreations mathematiques, four volumes Villars, Paris, 1891-1894 Reprinted by Albert Blanchard, Paris, 1960.(The Tower of Hanoi is discussed in volume 3, pages 55-59.)

Gauthier-209 R C Lyness, ‘Cycles,” The Mathematical Gazette 26 (1942), 62

210 R C Lyness, “Cycles,” The Mathematical Gazette 29 (1945), 231-233

211 Colin Maclaurin, Collected Letters, edited by Stella Mills Shiva lishing, Nantwich, Cheshire, 1982

Pub-212 P A MacMahon, “Application of a theory of permutations in circularprocession to the theory of numbers,” Proceedings of the London Math-ematical Society 23 (1892), 305-313

213 I^u V MatiiBsevich, “Diofantovost’ perechislimykh mnozhestv,” DokladyAkademii Nauk SSSR 191 (1970), 279-282 English translation, withamendments by the author, “Enumerable sets are diophantine,” SovietMathematics 11 (1970), 354-357

594 BIBLIOGRAPHY

214 Z.A Melzak, Compani’on to Concrete Mathematics Volume 1, Math- vi

ematical Techniques and Various Applications, Wiley, 1973; volume 2,

Mathematical Ideas, Modeling & Applications, Wiley, 1976

215 N S Mendelsohn, “Problem E 2227: Divisors of binomial coefficients,” 603.

American Mathematical Monthly 78 (1971), 201

216 W H Mills, “A prime representing function,” Bulletin of the American 603.

Mathematical Society, series 2, 53 (1947), 604

217 A Moessner, “Eine Bemerkung iiber die Potenzen der natiirlichen 604.

Zahlen,” Sitzungsberichte der Mathematisch - Naturwissenschaftliche

Klasse der Bayerischen Akademie der Wissenschaften, 1951, Heft 3, 29

218 Peter L Montgomery, “Problem E2686: LCM of binomial coefficients,” 603

American Mathematical Monthly 86 (1979), 131

219 Leo Moser, “Problem Es-6: Some reflections,” Fibonacci Quarterly 1,4 ~77

(1963), 75-76

220 T S Motzkin and E G Straus, “Some combinatorial extremum prob- 539

lems,” Proceedings of the American Mathematical Society 7 (1956),

1014-1021

221 C J Mozzochi, “On the difference between consecutive primes,” Journal 510

of Number Theory 24 (1986), 181-187

222 B R Myers, “Problem 5795: The spanning trees of an n-wheel,” Amer- 604.

ican Mathematical Monthly 79 (1972), 914-915

223 Isaac Newton, letter to John Collins (18 February 1670), in The Corre- 263.

spondence of Isaac Newton, volume 1, 27 Excerpted in The

Mathemat-ical Papers of Isaac Newton, volume 3, 563

224 Ivan Niven, Diophantine Approximations Interscience, 1963 602.

225 Ivan Niven, “Formal power series,” American Mathematical Monthly 76 318

(1969), 871-889

226 Blaise Pascal, “De numeris multiplicibus,” presented to AcadCmie Parisi- 602.

enne in 1654 and published with his Trait6 du triangle arithmbtique [227]

Reprinted in @uvres de Blaise Pascal, volume 3, 314-339

227 Blaise Pascal, “TraitC du triangle arithmetique,” in his TraitC du Triangle 155, 156, 594.Arithmetique, avec quelques autres petits traitez sur la mesme matiere,

Paris, 1665 Reprinted in C!Xuvres de Blaise Pascal (Hachette, 1904-1914),

volume 3, 445-503; Latin editions from 1654 in volume 11, 366-390

228 G P Patil, “On the evaluation of the negative binomial distribution with 605.

examples,” Technometrics 3 (1960) 501-505