% Bugs (sigh) in Concrete Math

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  Jan\or Feb\or Mar\or Apr\or May\or Jun\or
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\def\cutpar{{\parfillskip=0pt\par}}

\beginchapter {} Son of Concrete Errata

THIS IS A LIST
\g``And if you happen in reading to finde any more faultes not here
mentioned, as peraduenture you may, \dots\
I trust you will therefore impute no blame either vnto me or to the
Printer, but gently amend and correct them, accepting our good minde,
which was to haue had the booke passed to your handes vtterly
without fault, as touching the Printing.''\par
\hfill\dash---from the first\par
\hfill English edition\par
\hfill of Euclid [80]\g
 of all corrections made to the second printing (December 1988) of
{\sl Concrete Mathematics\/} by Graham, Knuth, and Patashnik.
The third printing (May 1989) incorporates these changes,
so the book is now perfect.

But you may disagree.
To claim a reward for any anomalies you spot that aren't shown here,
please send your comments to D.~E. Knuth, Computer Science Department,
Stanford University, Stanford CA 94305, as soon as possible.

\def\bugonpage#1(#2) \par{\bigbreak
  \vbox{\hrule width\hsize}
  \line{\lower3.5pt\vbox to13pt{}Page #1\hfil(#2)}%
  \nobreak\vskip-5pt
  \vbox{\hrule width\hsize}
  \nobreak\medskip}

\def\bib#1 {\noindent\hbox to\parindent{\bf#1\hss}\hangindent\parindent}

\bugonpage 3, lines 9 and 10 from the bottom (2 Jan 89)

\noindent
when $n$ has its smallest value, $n_0$; this is called the {\it basis}.
Then we prove
the statement for $n>n_0$, assuming that it has already been proved
for all values\cutpar

\bugonpage 23, line 5 from the bottom (13 Feb 89)

\noindent
$k$ that can be either true or false.)
 For the time being, we'll
assume that\cutpar

\bugonpage 60, lines 14--17 (8 Apr 89)

\noindent
the sum $\sum_{k\in K}a_k$ is said to
{\it "converge absolutely"\/} to the value $A=A^+-A^-$. If $A^+=\infty$
but $A^-$ is finite, the sum $\sum_{k\in K}a_k$ is said to {\it "diverge"\/}
to $+\infty$. Similarly, if $A^-=\infty$
but $A^+$ is finite, $\sum_{k\in K}a_k$ is said to diverge to $-\infty$.
If $A^+=A^-=\infty$, all bets are off.

\bugonpage 105, line 4 (8 Apr 89)

\def\chapno{4}\eqcount=8
\vskip-20pt
\begindisplay
\sum_{m\divides n}\,\sum_{k\divides m}a_{k,m}=
\sum_{k\divides n}\,\sum_{l\divides(n/k)}a_{k,kl}\,.
\eqno
\enddisplay

\bugonpage 105, line 15 (8 Apr 89)

\vskip-20pt
\begindisplay
\sum_{j,m}\,\sum_{k,l>0}a_{k,kl}\[n=jk]\[n/k=ml]=
\sum_m\sum_{k,l>0}a_{k,kl}\[n=mlk]\,,
\enddisplay

\bugonpage 106, line 2 (13 Feb 89)

\def\chapno{4}\eqcount=9
\vskip-20pt
\begindisplay
n=p_1\ldots p_m=\prod_{k=1}^m p_k\,,\qquad p_1\le\cdots\le p_m\,.
\eqno
\enddisplay

\bugonpage 112, line 12 (3 Mar 89)

\noindent
its largest value at $k=\half(n+1)$. Therefore

\bugonpage 121, bottom two lines (8 Apr 89)

\vskip-20pt
\begindisplay \openup6pt
&{m\over n}=f(RS)&\qquad\iff\qquad{m-n\over n}=f(S)\,,&\qquad\hbox{when $m>n$};\cr
&{m\over n}=f(LS)&\qquad\iff\qquad{m\over n-m}=f(S)\,,&\qquad\hbox{when $m<n$}.\cr
\enddisplay

\bugonpage 143, lines 9 and 10 (5 Dec 88)

\vskip-20pt
\begindisplay
n^{p^3}
	&= (n^p + p^2Q)^p\cr
	&= n^{p^2} \,+\, (p^2Q)^1 n^{p(p-1)} {p \choose 1}
		\,+\, (p^2Q)^2 n^{p(p-2)} {p \choose 2} \,+\, \cdots\cr
	&= n^{p^2} \,+\, p^3\hbox{\eulerbf Q}
\enddisplay

\bugonpage 156, quotation in margin (10 Dec 88)

\noindent
\g``C'est une chose estrange combien il est fertile en proprietez.''\g

\bugonpage 158, lines 9 and 10 (13 Feb 89)

\noindent
whenever $r$ is a positive
integer; so these two polynomials agree at infinitely many points, and
they must be identically equal.

\bugonpage 223, line 11 (10 Feb 89)

\noindent[Delete this line; a computer program by Wilf and Zeilberger
has just proved the identity by induction! Also, we could easily
guess the right-hand side if we factored the answers for small $n$.
Some day I'll insert a reference to the Wilf-Zeilberger paper, where
they show that the sum over $0\le k\le n$ turns out to equal
$-1/2$ times the right-hand side of \equ(5.113) when $n\=3$ \tmod3.]

\bugonpage 232, top line (3 Mar 89)

\item{18}
Find an alternative form analogous to \equ(5.35) for the product

\bugonpage 243, line 5 from the bottom (16 Mar 89)

Tables 244 and 245 show what
$n\brace k$ and~$@n@\brack k$ look like when $n$ and~$k$ are\cutpar

\bugonpage 273, line 12 (3 Jan 89)

\vskip-20pt
\begindisplay
{\sin z+x\cos z\over\cos z-x\sin z}
&=x+(1{+}x^2)z+(2x^3{+}2x){z^2\over2}+(6x^4{+}8x^2{+}2){z^3\over6}+\cdots\cr
\enddisplay

\bugonpage 275, lines 3--6 (13 Feb 89)

\vskip-20pt
\begindisplay \let\displaystyle=\textstyle \openup3pt
S_3(n)&=n^2(n-1)^2\!/4\,;\cr
S_4(n)&=n(n{-}\half)(n{-}1)(n-\half+\sqrt{7/12}\,)(n-\half-\sqrt{7/12}\,)/5\,;\cr
S_5(n)&=n^2(n-1)^2(n-\half+\sqrt{3/4}\,)(n-\half-\sqrt{3/4}\,)/6\,;\cr
S_6(n)&=n(n{-}\half)(n{-}1)(n{-}\half+\alpha)(n{-}\half-\alpha)
 (n{-}\half+\overline\alpha)(n{-}\half-\overline\alpha)\,,\cr
\enddisplay

\bugonpage 297, line 2 (8 Apr 89)

\vskip-20pt
\begindisplay
A_{n,0}&=a_n\,\[n\ge0]\,;\qquad A_{0,k}=0\,,\qquad\hbox{if $k>0@$};\cr
\enddisplay

\bugonpage 304, line 6 (13 Dec 88)

\indent
function $\sum_{n\ge1}z^{\lfloor n\alpha\rfloor}$, when
$\alpha$ is any positive irrational number.

\bugonpage 304, lines 6 and 7 from the bottom (6 Dec 88)

\item{86}%
What is the best way to extend the definition of $n\brace k$ to arbitrary
real values of $n$ and~$k$?

\bugonpage 325, line 10 from the bottom (9 Dec 88)

\vskip-20pt
\begindisplay
\phantom{Q^{\rm R}(z)}\qquad\iff
 Q(z)=q_0(1-\rho_1z)\ldots(1-\rho_mz)\,.
\enddisplay

\bugonpage 353, line 10 (2 Jan 89)

\vskip-20pt
\begindisplay
\phantom{\hat S(z)}
=\Bigl(B_0@{z^0\over0!}+B_1{z^1\over1!}+B_2{z^2\over2!}+\cdots\,\Bigr)
 \Bigl(n@{z^0\over1!}+n^2{z^1\over2!}+n^3{z^2\over3!}+\cdots\,\Bigr)\,.
\enddisplay


\bugonpage 353, line 10 from the bottom (3 Dec 88)

\vskip-20pt
\begindisplay
S_2(n)&=2!\,\Bigl(B_0@{n^3\over3!\,0!}+B_1@{n^2\over2!\,1!}
 +B_2@{n\over1!\,2!}\Bigr)
 &=\textstyle{1\over3}n^3-\half n^2+{1\over6}n\,.\cr
\enddisplay

\bugonpage 366, bottom two lines (16 Jan 89)

\item{}all the coefficients of $G(z)^2$ are less than some constant~$M$, then
infinitely many of the coefficients of $G(z)^2$ are zero.

\bugonpage 376, lines 7 and 8 (3 Mar 89)

\noindent
does it, but why? The main reason is
\g If he proved it in 1867, it's a classic '67 Chebyshev.\g
{\it "Chebyshev's inequality"\/} ([24$'@$] and [50$'@$]),
which states that the variance has a significant property:

\bugonpage 391, marginal quotation (25 Nov 88)

\g``\thinspace`You really are an\g

\bugonpage 414, line 9 (8 Apr 89)

\item{13}If
$X_1$, \dots, $X_{2n}$ are independent random variables with the same
distribu-\cutpar

\bugonpage 436, new marginal note for bottom of page (10 Dec 88)

\noindent
\g\vskip-20pt
(Note: The formula\/ $O(f(n))^2$ does not denote the set of all functions\/
$g(n)^2$ where\/ $g(n)$ is in $O(f(n))$; such functions $g(n)^2$ cannot
be negative, but the set\/ $O(f(n))^2$ includes negative functions. In general,
when $S$~is a set the notation\/ $S^2$ stands for the set of all products\/
$s_1s_2$ with $s_1$~and\/~$s_2$ in~$S$, not for the set of all
squares\/ $s^2$ with $s\in S$.)\g

\bugonpage 436, bottom line (10 Dec 88)

\noindent
use of ``exponents outside
the~$O@$'' to constant, positive integer exponents.

\bugonpage 437, line 14 (29 Nov 88)

\vskip-20pt
\begindisplay
S(z)=\sum_{0\le k<m}a_kz^k+z^m\sum_{n\ge m}a_nz^{n-m}
\enddisplay

\bugonpage 438, line 3 (13 Feb 89)

\def\chapno{9}\eqcount=28
\vskip-20pt
\begindisplay
n!&=\sqrt{2\pi n}\,\Bigl({n\over e}\Bigr)^{\!n}\biggl(1+{1\over12n}
 +{1\over288n^2}-{139\over51840n^3}+O\Bigl({1\over n^4}\Bigr)\biggr)\,.
 \eqno
\enddisplay

\bugonpage 438, line 9 from the bottom (13 Feb 89)

\vskip-20pt
\begindisplay
\sqrt{2\pi n}\,\Bigl({n\over e}\Bigr)^{\!n}\biggl(1+{1\over12n}
 +{1\over288n^2}-{139\over51840n^3}\biggr)\bigl(1+O(n^{-4})\bigr)
\enddisplay

\bugonpage 440, line 4 from the bottom (26 Nov 88)

\vskip-20pt
\begindisplay
O\bigl((n-1)^{-3}\bigr)&=O\bigl(n^{-3}(1-n^{-1})^{-3}\bigr)=O(n^{-3})\,.\cr
\enddisplay

\bugonpage 443, lines 5 and 6 (8 Apr 89)

\noindent
in fact, with this new estimate in hand we can conclude that
$\ln\ln p=\ln\ln n+O(\log\log n/\!@\log n)$, and \equ(9.45) now yields

\bugonpage 447, line 7 from the bottom (29 Nov 88)

\noindent
and the series converges when $x\ge1$ so we can truncate it at any
desired point.\cutpar

\bugonpage 452, line 11 from the bottom (3 Mar 89)

Those derivations were special cases of an important
\g\def@{\mskip-2mu}(This important method was pioneered by "Laplace"~[195$'@$].)\g
three-step asymp-\cutpar

\bugonpage 455, line 9 (8 Apr 89)

\def\chapno{9}\eqcount=65
\vskip-20pt
\begindisplay
\sum_{k\ge0}{\ln(n+2^k)\over k!}=e\ln n+\sum_{k=1}^{m-1}(-1)^{k+1}
{e^{2^k}\over k@n^k}+O\Bigl({1\over n^m}\Bigr)\,,
\eqno
\enddisplay
\bugonpage 460, lines 5 and 6 (13 Dec 88)

\noindent
Therefore its integral, $B_{4k+2}(x)/(4k+2)$, decreases for
$0<x<\half$ and increases for $\half<x<1$. Moreover, we have

\bugonpage 460, line 11 (13 Dec 88)

\noindent
$B_{4k+2}>0$. The integral of $B_{4k+2}(x)$ is
$B_{4k+3}(x)/(4k+3)$, which must therefore\cutpar

\bugonpage 463, line 11 (12 Dec 88)

\vskip-20pt
\begindisplay
&\qquad=\ln2+{39\over128}-n^{-1}+R_4(n)+O(n^{-5})\,.
\enddisplay

\bugonpage 468, line 9 (28 Nov 88)

\vskip-20pt
\begindisplay
\phantom{\sum\,}
F_m(\alpha,x)&=(x+\alpha)\ln(x+\alpha)- x+{\ln(x+\alpha)\over2}\cr
\enddisplay

\bugonpage 476, line 3 (25 Apr 89)

\item{10}%
Prove or disprove: $\cos O(x)=
1+O(x^2)$ for all real $x$.

\bugonpage 476, line 9 (7 Dec 88)

\item{14}Prove that $(n+\alpha)^{n+\beta}=n^{n+\beta}e^\alpha
\bigl(1+\alpha(\beta-\half\alpha)
n^{-1}+O(n^{-2})\bigr)$.

\bugonpage 481, line 4 from the bottom (8 Apr 89)

\noindent
[Move this line to the top of page 482; exercise 64 now becomes a Bonus
Problem, not a Research Problem.]

\parindent=30pt % for the Answers

\bugonpage 486, line 8 (3 Mar 89)

\noindent
the lines from $(n^{2k},0)$, and that all these intersection
points are distinct.

\bugonpage 489, lines 19 and 20 of the second printing (18 Apr 89)

\noindent
hence $2T_n=n+1-S_n$ and we have $T_n=\half\bigl(n+
\[\hbox{$n$ is odd}]\bigr)$.
Finally, the same approach yields

\bugonpage 491, line 9 (18 Apr 89)

\noindent
$n(x{-}n)=(x{-}1)+(x{-}3)+\cdots+(x{-}2n{+}1)$. If $2n-1\le x<2n$
they are,\cutpar

\bugonpage 510, lines 11 and 12 from the bottom (3 Mar 89)

\textindent{4.67}%
M. "Szegedy" has proved this conjecture for all large~$n$;
see [284$'@$], [77, pp.~78--79], and~[49].

\bugonpage 514, lines 6--8 (2 Jan 89)

"Gauss" [116, \S7]
derived analogous relations
between $F(a,b;c;z)$ and any two ``"contiguous"''
hypergeometrics in which a parameter has been changed by $\pm1$.
Rainville [242$'@$] generalized this to cases with more parameters.

\bugonpage 527, line 6 from the bottom (8 Apr 89)

\textindent{6.16}%
 $A_{n,k}=\sum_{j\ge0}a_j{n-j\brace k}$; this sum is always finite.

\bugonpage 529, line 9 (21 Dec 88)

\textindent{6.31}%
 Since $x\_^n=(x+n-1)\_n=\sum_k{n\choose k}x\_k(n-1)\_{n-k}$, we have
\def\\{\atopwithdelims\vert\vert}%
${n\\k}={n\choose k}{(n-1)}\_{n-k}$. These coefficients, incidentally, satisfy
the recurrence

\bugonpage 556, margin (4 Jan 89)

\noindent
\g ``"Toto", I have a feeling we're not in Kansas anymore.''\g

\bugonpage 560, line 4 from the bottom (24 Jan 89)

\vskip-20pt
\begindisplay
\qquad={2n+1\over 2^{2n}}{2n\choose n}-1\,.
\enddisplay

\bugonpage 567, lines 7--9 (25 Apr 89)

\textindent{9.10}%
 If $g(x)$ belongs to the left, so that $g(x)=\cos y$ for some $y$,
where $\vert y\vert\le C\vert x\vert$ for some $C$, then $0\le1-g(x)
=2\sin^2(y/2)\le\half y^2\le\half C^2 x^2$;
hence the set on the left is contained in the set
on the right, and the formula is true.

\bugonpage 567, lines 11 and 12 (10 Dec 88)

\noindent Thus $(x+y)^2=O(x^2)+O(y^2)$. Thus
$O(x+y)^2=O\bigl((x+y)^2\bigr)=
O\bigl(O(x^2)+O(y^2)\bigr)=O\bigl(O(x^2)\bigr)+O\bigl(O(y^2)\bigr)
=O(x^2)+O(y^2)$.

\bugonpage 576, line 8 from the bottom (13 Jan 89)

\noindent
by exercise 5.22. Hence the result
follows from exercises 6.64 and 9.44.

\bugonpage 577, lines 9--13 from the bottom (8 Apr 89)

\noindent ercise~58 by
induction on~$m$.) If $0<x<1$, the integral $\int_x^{1/2}\sin N\pi t\,dt/
\!\sin\pi t$ can be expressed as a sum of $N$ integrals that are each $O(N^{-2})$,
so it is $O(N^{-1})$; the constant implied by this~$O$ may depend on~$x$.
Integrating the identity $\sum_{n=1}^N\cos2n\pi t=
\Re\bigl(e^{2\pi it}(e^{2N\pi it}-1)/(e^{2\pi it}-1)\bigr)
=-\half+\half\sin(2N+1)\pi t
/\!\sin\pi t$ and letting $N\to\infty$ now gives $\sum_{n\ge1}(\sin2n\pi x)/n
={\pi\over2}-\pi x$, a relation that "Euler" once stated without proof~%
[85$@'$]. Integrate again to get the desired formula. (This solution
was suggested by E.\thinspace M.\thinspace E. "Wermuth".)

\let\,=\thinspace \frenchspacing
\parindent=22pt \tolerance=1000 % for the Bibliography

\bugonpage 579, lines 14 and 15 from the bottom (10 Dec 88)

\bib18
Samuel "Beatty", ``Problem 3177,'' {\sl American Mathematical Monthly\/
\bf 34} (1927), 159--160.

\bugonpage 579, line 7 from the bottom (8 Dec 88)

\bib22
Jacobi Bernoulli, {\sl Ars Conjectandi}, opus posthumum.
Basel, 1713. Re-\cutpar

\bugonpage 580, new entry (3 Mar 89)

\bib{24$\kern-1pt'$}
\g376.\g
J. "Bienaym\'e", ``Consid\'erations \`a l'appui de la d\'ecouverte de "Laplace"
sur la loi de probabilit\'e dans la m\'ethode des moindres carr\'es,''
{\sl Comptes Rendus hebdomadaires des s\'eances de l'Acad\'emie des Sciences\/}
(Paris) {\bf37} (1853), 309--324.

\bugonpage 581, second line of reference 50 (3 Mar 89)

\bib{}
{\sl nal de Math\'ematiques pures et appliqu\'ees\/ \bf17}
(1852), 366--390. Reprinted in his {\sl \OE uvres}, volume~1, 51--70.

\bugonpage 581, new entry (3 Mar 89)

\bib{50$\kern-1pt'$}
\g376.\g
P.\,L. "Chebyshev", ``O srednikh velichinakh,''
{\sl Matematicheski\u\i\ Sbornik'\/ \bf2} (1867), 1--9. Reprinted in his
{\sl Polnoe Sobranie Sochineni\u\i}, volume~2, 431--437. \ French translation,
``Des valeurs moyennes,'' {\sl Journal de Math\'e\-ma\-tiques
pures et appliqu\'ees}, series 2, {\bf12} (1867), 177--184; reprinted
in his {\sl \OE uvres}, volume~1, 685--694.

\bugonpage 582, line 3 from the bottom (23 Nov 88)

der
Zahlen,'' {\sl Bericht \"uber die Verhandlungen der K\"oniglich-Preu\ss ischen%
\cutpar}

\bugonpage 583, line 3 from the bottom (10 Dec 88)

\bib80
"Euclid", {\sl \char6TOI\kern.5ptXEIA}.
 \ Ancient manuscript first printed in Basel, 1533.\cutpar

\bugonpage 584, line 2 (8 Apr 89)

{\sl respondance Math\'ematique et Physique de Quelques C\'el\`ebres
G\'eom\`etres}\cutpar

\bugonpage 584, new entry (8 Apr 89)

\bib{85$\kern-1pt'$}
\g577.\g
Leonhard "Euler", letter to Christian "Goldbach" (4~July 1744), in
{\sl Correspondance Math\'ematique et Physique de Quelques C\'el\`ebres
G\'eom\`etres du XVIII\kern1pt\raise.2ex\hbox{\`eme} Si\`ecle},
 edited by P.\,H. "Fuss", St.~Petersburg, 1843, volume~1, 278--293.

\bugonpage 584, line 18 (23 Nov 88)

Lausanne, 1748.
Reprinted in his {\sl Opera Omnia}, series~1, volume 8.\cutpar

\bugonpage 584, line 24 (23 Nov 88)

\bib88
Leonhard "Euler", {\sl Institutiones Calculi Differentialis cum eius usu
in An-\cutpar}

\bugonpage 585, lines 6--9 (23 Nov 88)

\bib93
Leonhard "Euler", ``Observationes circa bina biquadrata quorum summam in duo
alia biquadrata resolvere liceat,''
{\sl Novi commentarii academi{\ae} scientiarum Petropolitan{\ae}
\bf17} (1772), 64--69.
Reprinted in his {\sl Opera Omnia}, series~1, volume~3, 211--217.

\bugonpage 588, line 12 from the bottom (23 Nov 88)

\bib143
G.\,H. "Halphen", ``Sur des suites de fractions analogues \`a la suite de~Farey,''
{\sl Bulletin de la Soci\'et\'e math\'ematique de France\/ \bf5} (1876),
170--175. Reprinted in his {\sl \OE uvres}, volume~2, 102--107.

\bugonpage 591, line 23 (10 Dec 88)

\bib186
Christian "Kramp", {\sl \'El\'emens d'arithm\'etique universelle}. \
Cologne, 1808.

\bugonpage 592, lines 9 and 10 (22 Dec 88)

\bib192
J.-L. de la Grange ["Lagrange"], ``Sur une nouvelle esp\`ece de calcul
 r\'elatif % sic, in the original
\`a la diff\'erentia\-tion \& \`a l'int\'egration des quantit\'es variables,''
{\sl Nouveaux\cutpar}

\bugonpage 592, new entry (3 Mar 89)

\bib{195$\kern-1pt'$}
\g452.\g
P.\thinspace S. de la Place ["Laplace"], ``M\'emoire sur les approximations
des Formules
qui sont fonctions de tr\`es-grands nombres,'' {\sl M\'emoires de l'Academie
royale des Sciences de Paris\/} (1782), 1--88. Reprinted in his
{\sl \OE uvres Compl\`etes\/ \bf10}, 207--291.

\bugonpage 594, lines 16--18 (23 Nov 88)

\bib226
Blaise "Pascal", ``De numeris multiplicibus,'' presented to Acad\'emie
 Pari\-si\-enne in 1654 and published with his {\sl Trait\'e du
triangle arithm\'etique\/}~\kern-.2pt[227].
 Reprinted in {\sl \OE uvres de Blaise Pascal}, volume~3, 314--339.

\bugonpage 594, line 14 from the bottom (23 Nov 88)

\bib{}
Paris, 1665. Reprinted
in {\sl \OE uvres de Blaise Pascal\/} (Hachette, 1904--1914), volume~3, 445--503;
Latin editions from 1654 in volume~11, 366--390.

\bugonpage 595, new entry (2 Jan 89)

\bib{242$\kern-1pt'$}
\g514.\g
Earl D. Rainville, ``The contiguous function relations for $_pF_q$ with
applications to Bateman's $J_n^{u,v}$ and Rice's $H_n(\zeta,p,v)$,''
{\sl Bulletin of the American Mathematical Society}, series~2,
{\bf51} (1945), 714--723.

\bugonpage 596, line 24 (25 Nov 88)

{\sl Mathematical Monthly\/ \bf94} (1987), 36--46.

\bugonpage 598, new entry (3 Mar 89)

\bib{284$\kern-1pt'$}
\g510.\g
M. "Szegedy", ``The solution of "Graham"'s greatest common divisor
problem,'' {\sl Combinatorica\/ \bf6} (1986), 67--71.


\parindent=30pt % for the Credits

\bugonpage 603, right column (21 Dec 88)

\textindent{6.31}Lah [193]; R.\thinspace W. Floyd.*

\parindent=0pt % for the Index

\bugonpage 607, left column (24 Apr 89)

Bateman, Harry, 595.

\bugonpage 607, right column (3 Mar 89)

Bienaym\'e, Ir\'en\'ee Jules, 580.

\bugonpage 610, left column (9 Mar 89)

Divergent sums, 60, 334, 517.

\bugonpage 611, right column (9 Mar 89)

Factorial function, 111--115, 332--334.

\bugonpage 612, left column (21 Dec 88)

Floyd, Robert W, 603, 604.

\bugonpage 613, second line of entry for R.\thinspace L. Graham (3 Mar 89)

\hskip20pt 492, 582, 583, 587--588, 598, 601, 602.

\bugonpage 615, right column (3 Mar 89)

Laplace, Pierre Simon, marquis de, 452, 580, 592.

\bugonpage 616, left column (3 Mar 89)

\hangindent=20pt
Mathews, Edwin Lee (= 41), 8, 21, 94, 105,\hfil\break 106, 329.

\bugonpage 619, new entry (2 Jan 89)

Rainville, Earl David, 514, 595.

\bugonpage 619, another new entry (24 Apr 89)

Rice, Stephan Oswald, 595.

\bugonpage 620, left column (3 Mar 89)

\hangindent=20pt
Seaver, George Thomas (= 41), 8, 21, 94, 105,\hfil\break 106, 329.

\bugonpage 621, right column (3 Mar 89)

Szegedy, M\'ari\'o, 510, 581, 598.

\bugonpage 623, new entry (8 Apr 89)

Wermuth, Edgar M.\thinspace E., 577.

\bye
