% CHANGES TO VOLUME 2 OF THE ART OF COMPUTER PROGRAMMING
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% Copyright (C) 2022,2023,2024, 2025 by Donald E. Knuth
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\def\date#1.#2.#3.{% convert "yy.mm.dd." to "dd Mon 19yy"
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 \ \ifnum #1<97 \hundred#1\else19#1\fi} \def\hundred{20} % the "century" for dates before '97 \def\ex #1. [#2]{\ninepoint \textindent{\bf#1.}[{\it#2\/}]\kern6pt} \def\EX #1. [#2]{\ninepoint \textindent{\llap{\manfnt x}\bf#1.}[{\it#2\/}]\kern6pt} \def\foottext#1{\medskip \hrule height\ruleht width5pc \kern-\ruleht \kern3pt \eightpoint \smallskip\textindent{#1}} \def\volheadline#1{\line{\cleaders\hbox{\raise3pt\hbox{\manfnt\char'36}}\hfill \titlefont\ #1\ \cleaders\hbox{\raise3pt\hbox{\manfnt\char'36}}\hfill}} \def\refin#1 {\let&#124;\inref \input #1.ref \let&#124;\crossref} \let\defaultpointsize=\tenpoint %%%%%%%%%%%%%% opening remarks %%%%%%%%%%%%%%%%%%%%%%%% \def\lhead{INTRODUCTION} \let\rhead=\lhead \titlepage \volheadline{THE ART OF COMPUTER PROGRAMMING} \bigskip \volheadline{ERRATA TO VOLUME 2 (after 2021)} \bigskip \noindent This document is a transcript of the notes that I have been making in my personal copy of {\sl The Art of Computer Programming}, Volume~2 (third edition, 45th printing) since it was first printed in 2022. Previous errata are recorded in another file `{\tt all2-pre.ps}'. \ifall Four levels of updates\dash---``errors,'' ``amendments,'' ``plans,'' and ``improvements''\dash---appear, indicated by four \else Three levels of updates\dash---``errors,'' ``amendments,'' and ``plans''\dash---appear, indicated by three \fi different typographic conventions: \begingroup\def\hundred{17} \bugonpage 0.666 line 1 (76.07.04) Technical or typographical errors (aka bugs) are the most critical items, so they are flagged with a `\thinspace{\manfnt x}\thinspace' preceding the page number. The date on which I first was told about the bug is shown; this is the effective date on which I paid the finder's fee. The necessary corrections are indicated in a straightforward way. If,~for example, the book says `$n$' where it should have said `$n+1$', the change is shown thus: \smallskip $n$ \becomes $n+1$ \endchange \amendpage 0.666 line 2 (89.07.14) Amendments to the text appear in the same format as bugs, but without the~`\thinspace{\manfnt x}\thinspace'. These are things I wish I had known about or thought of when I wrote the original text, so I added them later. The date is the date I drafted the new text. \endchange \def\hundred{19} \planforpage 0.666 line 3 (17.11.20) Plans for the future represent a third kind of item. In such notes I~sketched my intentions about things that I wasn't ready to flesh out further when I~wrote them down. You can identify these items because they're written in slanted type, and preceded by a bunch of dots `\hbox to 6em{\leaders\hbox to 5pt{\hss.\hss}\hfill}' leading to the date on which I recorded the plan in my files. \endchange \improvepage 0.666 line 4 (38.01.10) The fourth and final category\dash---indicated by page and line number in smaller, slanted type\dash---consists of minor corrections or improvements that most readers don't want to know about, because they are so trivial. You wouldn't even be seeing these items if you hadn't specifically chosen to print the complete errata list in all its gory details. Are you sure you wanted to do that? \endchange \endgroup \ifall\else\medskip\ninepoint My personal file of updates also includes a fourth category of items, not shown in this list. They are miscellaneous minor corrections or improvements that most readers don't want to know about, because they are so trivial. If you really want to see all of the gory details, you can download the full list from Internet webpage $$\.{http://www-cs-faculty.stanford.edu/\char`\~knuth/taocp.html}$$ by selecting the ``long form'' of the errata. \fi \medskip \tenpoint My shelves at home are bursting with preprints and reprints of significant research results that I want to digest and summarize, where appropriate, in the ultimate edition of Volume~2. I didn't do that in the third edition because I would surely have to do it over again later: New results continue to pour forth at a great rate, and I will have time to rewrite that volume only~once. Volumes 4 and~5 need to be finished first. So I've put most of my effort so far into writing up those parts of the total picture that seem to have converged to their near-final form. It follows, somewhat paradoxically, that the updates in this document are most current in the areas where there has been least activity. On the other hand I do believe that the changes listed here bring Volume~2 completely up to date in two respects: (1)~All of the research problems in the previous edition\dash---i.e., all exercises that were rated 46 and above\dash---have received new ratings of 45 or less whenever I learned of a solution; and in such cases, the answer now refers to that solution. (2)~All of the historical information about pioneering developments has been amended whenever new details have come to my attention. \beginconstruction The ultimate, glorious, 100\% perfect editions of Volumes 1--4 are works in progress. Please let me know of any improvements that you think I ought to make. Send your comments either by snail mail to D.~E. Knuth, Computer Science, Gates Building 1B, Stanford University, Stanford CA~94305-9015, or by email to {\tt taocp{\char`\@}cs.stanford.edu}. (Use email for book suggestions only, please\dash---all other correspondence is returned unread to the sender, or discarded, because I have no time to read ordinary email.) Although I'm working full time on Volume~4 these days, I~will try to reply to all such messages within a year of receipt. Current news about {\sl The Art of Computer Programming\/} is posted on $$\.{http://www-cs-faculty.stanford.edu/\char`~knuth/taocp.html}$$ and updated regularly. \par\endconstruction \rightline{\dash---Don Knuth, January 2011} \bigskip \bigskip {\quoteformat Oh! If only someone would give me time, time, time to do everything\/ {\rm properly}, to read everything at\/ {\rm my own\/} tempo, to take it apart and put it together again. \author KARL BARTH (1922) % letter to Edvard Thurneysen, spring 1922 % found in Revolutionary Theology in the Making: Barth-Thurneysen % Correspondence 1914--1925, tr by Jas. D. Smart % Richmond VA: John Knox Press, 1964, p43 \vfill\eject } \def\today{\number\day\space\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\year} %%%%%%%%%%%%%%% CHANGES FOR VOLUME 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\lhead{CHANGES TO VOLUME 2: SEMINUMERICAL ALGORITHMS} \let\rhead=\lhead \titlepage \volheadline{SEMINUMERICAL ALGORITHMS} \bigskip \rightline{Copyright \copyright\ 2022, 2023, 2024, 2025 Addison\with Wesley} \rightline{Last updated \today} \bigskip \rightline{\sl Most of these corrections have already been made in recent printings.} \smallskip \let\defaultpointsize=\tenpoint \bugonpage 2.3 line 12 (24.09.23) {\sl Games, Gods, and Gambling\/} \becomes {\sl Games, Gods and Gambling\/} \endchange \amendpage 2.3 line $-8$ (24.09.23) square it to get \becomes square it to get a 20-digit product, \endchange \bugonpage 2.27 line 20 (24.12.23) middle square method \becomes middle-square method \endchange \amendpage 2.34 line 18 (24.06.18) obtaining \becomes and $m$ equal to the word size, obtaining \endchange \bugonpage 2.35 lines 11 and 12 (24.12.23) J. Todd and O. Taussky Todd \becomes O. Taussky and J. Todd \endchange \amendpage 2.39 line $-4$ (24.06.18) $\lambda\gg k$ \becomes $\lambda$ greatly exceeds $k$ \endchange \amendpage 2.57 line 3 (24.06.18) $\displaystyle \max_{-\infty<x<+\infty}$ \becomes $\displaystyle \sup_{-\infty<x<+\infty}$ \endchange \bugonpage 2.57 Smirnov reference near the bottom (22.11.10) 176--206 \becomes 179--206 \endchange \amendpage 2.61 line 7 (22.12.14) eleven kinds \becomes twelve kinds \endchange \improvepage 2.61 lines 19 and 26 (24.06.18) between zero and one \becomes in the half-open interval $[0\dts1)$\nl between 0 and $d-1$ \becomes in the half-open interval $[0\dts d)$ \endchange \improvepage 2.61 lines $-11$ and $-10$ (24.09.24) the value of $d$ will probably be different in different tests. \becomes $d$ will generally have different values, depending on the test that is being made. \endchange \amendpage 2.140 line 2 of exercise 25 (24.08.17) \ninepoint $\half$. \becomes $\half ,ドル and consider the bitwise operations $\bor$ and $\band$ (``or'' and ``and''). \endchange \amendpage 2.149 lines 20 and 21 (24.06.30) {\sl Probability, Statistics,} \dots\ 1957), \becomes {\sl Wahrscheinlichkeit Statistik und Wahrheit\/} (Vienna:\ Springer, 1928), \endchange \amendpage 2.155 lines 8 and 9 (24.06.30) of points $(U_{2n-1}, U_{2n})$ is deficient \dots\ other points $(U_{2n}, U_{2n+1})$ \becomes of points $(U_{2n}, U_{2n+1})$ is deficient in some area, the other points $(U_{2n+1}, U_{2n+2})$ \endchange \amendpage 2.172 addendum to the statement of Lemma P4 (24.06.30) . \becomes {\sl, where $T(f)$ and $T(g)$ are the running times needed to compute $f$ and $g$.} \endchange \bugonpage 2.182 replacement for lines 4 and 5 of exercise 26 (24.09.23) $$\vcenter{\halign{$(V_{2n},V_{2n+1})=(#)$&\qquad if $(U_{2n},U_{2n+1})#G,$\cr U_{2n},U_{2n+1}&\in\cr U_{2n+1},U_{2n}&\notin\cr}}$$ \endchange \improvepage 2.185 line 2 of point {(v)} (24.06.30) in common with $m$ when $m$ is the computer's word size. \becomes in common with $m$. \endchange \amendpage 2.190 new rating for exercise 5 (23.11.15) \ninepoint [{\it46\/}] \becomes [{\it45}] \endchange \amendpage 2.198 line 22 (23.11.08) 1586--1610. \becomes 1586--1610. (In India, al-K\=ash\ii's contemporary M\=adhava had computed 9ドル\cdot10^{11}\pi\approx 2827433388233,ドル according to N{\ii}lakantha's commentary on Ga\d{n}ita \S10 of the {\sl \=Aryabha\d{t}\={\i}ya\/}.) \endchange \amendpage 2.199 line 9 (23.11.08) man's custom \becomes human custom \endchange \amendpage 2.200 lines 13--14 (22.08.10) as early as 1679 \dots\ ``decimal''). \becomes already in 1679 or earlier, and that he often experimented with radix-16 numbers. \endchange \amendpage 2.200 line 17 (22.08.10) 1973). Zacher points out \becomes 1973), and extended by L.~Strickland and H. Lewis (Cambridge, Mass.:\ MIT Press, 2022), who have provided English translations. Zacher pointed out \endchange \bugonpage 2.200 line $-18$ (22.08.10) 35-digit \becomes 35-place \endchange \improvepage 2.200 lines $-13$ and $-12$ (22.08.10) The motive \dots\ could be observed in \becomes Bernoulli and Leibniz were hoping to discover a simple pattern in \endchange \amendpage 2.200 new paragraph just before ``Charles XII'' (24.04.12) A book about binary arithmetic, {\sl Arithmeticus Perfectus\/} by W. J. Pelicanus (Prague:~1712), $\rm xii+84$ pages, included examples of how one might extract square roots and cube roots in radix~2. This work\dash---``ahead of its time''\dash---was almost totally forgotten for more than 300 years. [See D.~Zlatopolski, {\sl Historia Mathematica\/ \bf62} (2023), 40--50.] \endchange \amendpage 2.201 line 1 (22.08.10) \indent More than a century later, a prominent \becomes\nl\indent The eccentric British schoolmaster T. W. Hill spent more than 50 years of his life experimenting with radix~16. His public lecture in 1845, published posthumously, described ingenious ways to say and to write hexadecimal numbers and fractions [{\sl Selections from the Papers of the Late Thomas Wright Hill, Esq.\ F.R.A.S.} (London:\ 1860), 63--85]. But he gave no examples of arithmetic. Independently, a prominent \endchange \amendpage 2.201 delete an obsolete paragraph (22.08.10) The book {\sl History of Binary\/} \dots\ 243]. \becomes \endchange \amendpage 2.202 line 5 (22.08.08) sexadecimal \becomes sexdecimal \endchange \amendpage 2.209 lines 25 and 26 (22.11.02) Several questions \dots\ by W. Parry, \becomes Intriguing questions concerning {\it irrational\/} radices have been investigated by A.~R\'enyi, {\sl Acta Math.\ Acad.\ Sci.\ Hung.\ \bf 8} (1957), 477--493; W.~Parry, \endchange \amendpage 2.209 just before the exercises (22.11.02) Gray code \becomes regular continued fractions (Section 4.5.3); Gray codes \endchange \amendpage 2.213 line 2 (23.12.18) {\sl Crelle\/ \bf127} (1904), 51--84. \becomes\nl {\sl Jahresbericht der Deutschen Mathematiker-Vereinigung\/ \bf6} (1897), 83--88. \endchange \bugonpage 2.226 line 17 (23.11.15) Reading, Mass. \becomes Cambridge, Mass. \endchange \amendpage 2.232 replacement for {\eq(15), \eq(16)} and the following reference (24.09.23) \noindent $$\vbox{\tabskip 0pt plus 1000pt \halign to \hsize{\hfill$\displaystyle{#}$\tabskip 0pt &$\displaystyle{\null#}$\hfill\qquad&\hfill$\displaystyle{#}$&$\displaystyle{\null#}$\hfill \tabskip 0pt plus 1000pt&\hfill$#$\tabskip0pt\cr M_0 &= 0,&M_k &= M_{k-1} \oplus (x_k \ominus M_{k-1}) \oslash k,&\eq(15)\cr \noalign{\vskip 4pt} S_0 &= 0,& S_k &= S_{k-1} \oplus (x_k \ominus M_{k-1}) \otimes (x_k \ominus M_k),&\eq(16)\cr}}$$ for 1ドル \le k \le n,ドル where $\sigma = \sqrt{S_n/(n - 1)}$. [See B. P. Welford, {\sl Technometrics\/ \bf 4} (1962), 419--420; A.~J. van Reeken, {\sl CACM\/ \bf11} (1968), 149--150.] \endchange \improvepage 2.245 line $-6$ of exercise 28 (25.02.12) precision $b^{@p}$ \becomes fraction size $b^{@p}$ \endchange \amendpage 2.263 line $-4$ of exercise 12 (24.06.18) \ninepoint max \becomes sup \endchange \bugonpage 2.311 in the Harvey / van der Hoeven reference (22.11.10) {\bf 191} \becomes (2) {\bf 193} \endchange \amendpage 2.324 line 10 (24.09.25) correct. \becomes correct. (For example, after three steps we could check that 6ドル+6+8\equiv4-3+2-1$ (modulo~9).) \endchange \bugonpage 3.328 last line of exercise 9 (24.07.29) \ninepoint $\lfloor v_k[u]/16\rfloor$ \becomes $\lfloor v_k(u)/16\rfloor$ \endchange \amendpage 2.330 the quote by Hofstadter (24.10.09) {\eightss (1983) \becomes (1985)} \endchange \improvepage 2.354 replacement for exercise 11 (24.07.29) \ninepoint \ex11. [M22] Let $u$ and $v$ be random positive integers. (a)~What is the probability that $\gcd(u, v) \le 3$? (See Theorem~D\null.) (b)~What is the {\it average\/} value of $\gcd(u, v)$? \endchange \improvepage 2.359 line 5 (24.07.29) pattern that appears \becomes patterns that appear \endchange \amendpage 2.409 changes to accommodate the new world record prime (24.12.10) line 7: add `136279841' at the end of list \eq(26)\nl line 11: But the remaining exponents \becomes The next 17\nl lines 15--16: personal computers \dots They used a program\becomes personal computers. And the biggest one was discovered in 2024 by Luke {Durant}\dash---who assembled a network of thousands of computers in 17 countries(!). They used programs\nl line $-18$: unknown Mersenne \dots might\becomes unknown Mersenne primes less than 2ドル^{136279841}-1$ might\nl lines $-17$ and $-16$: all exponents \dots volunteers \becomes all exponents $<70{,}832{,}851$ were checked as of December 10, 2024; his volunteers\nl line $-15$: Since \dots digits \becomes Since 2ドル^{136279841}-1$ has more than 41 million decimal digits \endchange \amendpage 2.409 lines $-17$ and $-16$ (25.04.01) all exponents \dots; his \becomes all exponents $<72{,}072{,}997$ have been checked as of March 31, 2025; his \endchange \bugonpage 2.423 line 3 of Lemma H (24.08.17) $u = c_1 \cdot v_1(x)$ \becomes $u(x) = c_1 \cdot v_1(x)$ \endchange \bugonpage 2.431 in Table 1 (22.11.21) rows $C_1$ and $C_0$: $d_2^2b_6^4/c_4^5$ \becomes $d_2^3b_6^4/c_4^5$\nl row $D_0$: $e_2^2c_4^2/d_2^3b_6^2$ \becomes $e_1^2c_4^2/d_2^3b_6^2$ \endchange \bugonpage 2.432 three lines before {\eq(23)} (22.12.08) $C_{n_2+n_j-1}$ becomes $C_{n_2-n_j-1}$ \endchange \improvepage 2.465 in {\eq(2)} (24.08.17) $k\le j<i$ \becomes 0ドル\le k\le j<i$ \endchange \amendpage 2.477 lines 20 and 21 (22.08.12) using a backtrack \dots cases. \becomes using the backtrack method in a program called \.{ACHAIN0} now on the Internet, involved a detailed examination of more than 100 cases. \endchange \improvepage 2.477 just before the table of {$c(r)$} (24.09.24) The computation of $l(n)$ \becomes The proof that $l(n)>r-1$
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\amendpage 2.477 replacement for the table at the bottom (22.08.13)
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%\bugonpage 2.478 line 6 (11.08.30)
%$c(33)$ \becomes $c(39)$
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\amendpage 2.478 lines 4--8 (22.08.13)
The values listed \dots\ this agrees \becomes
The values of $c(r)$ listed here for 18ドル<r\le28$ were discovered by Achim Flammenkamp before 1998, except that $c(24)$ was first computed by Daniel Bleichenbacher; $c(29)$ through $c(47)$ are due to Neill Clift, between 2004 and 2021. Notice that the formula 1ドル+r/(1+1/\!@\lg r)$ gives a fairly good approximation to $\lg c(r)$ for 10ドル< r\le47$; this agrees \endchange \amendpage 2.478 three lines before {\eq(49)} (22.08.13) large $r$. \becomes large $r$. (Empirically, $d(r)\approx 1.7c(r)$ for 35ドル\le r\le44$.) \endchange \amendpage 2.478 lines $-11$ through $-5$ (24.09.24) Computer calculations \dots~Arnold Scholz \becomes\nl Notice that 2ドル^n-1$ is the worst case for the binary method, because $\nu(2^n-1)=n$. E.~G. Thurber [{\sl Discrete Math.\ \bf 16} (1976), 279--289] has shown that several of these values, including the case $n = 32,ドル can actually be calculated by hand. Extensive computer calculations by Neill Clift [{\sl Computing\/ \bf91} (2011), 265--284; also Thurber and Clift, {\sl Discrete Math.\ \bf344} (2021), \#112200, 15~pp.] show that $l(2^n-1)$ is in fact exactly {\it equal\/} to %$n-1+l(n)$ for 1ドル\le n\le64$. % but he doesn't think it will hold in general $n-1+l(n)$ whenever $l(n)\le9$. % also n=127,139,141 (his letter of 22Feb2022) But Clift discovered in 2024 that equality {\it fails\/} when $n=1541527$! Arnold Scholz \endchange \amendpage 2.479 bottom line (24.09.24) (see exercise 42). \dots\ in doubt. \becomes and for many larger values of~$n$. (See exercise 42 and its answer.) \endchange \amendpage 2.485 rewording exercise 37 (22.08.14) \ninepoint contains $m$ given \dots\ \eq(25). \becomes contains each of the numbers $n_1,ドル \dots, $n_m$. Generalizing \eq(25), prove that $l(n_1,\ldots,n_m)\le \lambda (n)+m\lambda (n)/\lambda \lambda (n)+ O\bigl( \lambda (n)@\lambda \lambda \lambda (n)/\lambda \lambda (n)^2 \bigr),ドル $n=\max\{n_1,\ldots,n_m\}$. \endchange \amendpage 2.485 in exercise 43 (24.09.24) \ninepoint Does equality always~hold? \becomes \endchange \bugonpage 2.505 line $-2$ (25.07.17) MacMillan \becomes Macmillan \endchange \amendpage 2.537 new exercise (24.04.01) \EX29. [\HM21] (Ira M. Gessel.) Every bivariate generating function corresponds naturally to a sequence of $n\times n$ matrices, when we let $$A(w,z)=\sum_{m=0}^\infty\sum_{n=0}^\infty a_{m,n},円w^mz^n\quad \longleftrightarrow\quad A_n=\pmatrix{a_{0,0}&a_{0,1}&\ldots&a_{0,n-1}\cr a_{1,0}&a_{1,1}&\ldots&a_{1,n-1}\cr \vdots&\vdots&\ddots&\vdots\cr a_{n-1,0}&a_{n-1,1}&\ldots&a_{n-1,n-1}\cr }.$$ \item{a)} Suppose $T(w,z)=p(w)@@q(z)A\bigl(w@r(w),z@s(z)\bigr),ドル where $p,ドル $q,ドル $r,ドル and $s$ are arbitrary power series. If the corresponding $n\times n$ matrices are denoted by $T_n,ドル prove that $$\det T_n\;=\;p(0)^nq(0)^nr(0)^{n(n+1)/2}s(0)^{n(n+1)/2}\det A_n.$$ \item{b)} Use this technique to evaluate $\det A_n$ when $A(w,z)=e^{w+z}\!/(e^w+e^z-e^{w+z})$. \endchange \bugonpage 2.537 in the Stillingfleet quote (24.05.24) {\eightssi which first\/} \becomes {\eightssi which at first} \endchange \let\defaultpointsize=\ninepoint % get ready for answer pages \amendpage 2.538 new answer for Notes on the Exercises (23.11.24) \ans2. You can check whether you understand something, by looking at the answer. \endchange \bugonpage 2.539 line 10 of answer 6 (24.12.23) {\sl Sitzungsberichte\/} \dots 82--83 \becomes\nl {\sl Sitzungsberichte K\"onigl.\ preu\ss.\ Akad.\ Wiss.\ }(Berlin:~1895), 164--165 \endchange \amendpage 2.543 line 2 of answer 3 (24.12.23) $m/d$ values. \becomes $m/d$ values, modulo~$m$. \endchange \amendpage 2.545 replacement for lines $-5$ and $-4$ of answer 11 (24.12.23) \noindent $$\eqalign{ &t\gets 70317(x\mod30540)-2467\lfloor x/30540\rfloor,,円\cr &t\gets 883(t\mod2432031)-274\lfloor t/2432031\rfloor,円.\cr}$$ \endchange \bugonpage 2.552 line 1 of answer 7 (24.12.23) (1897) \becomes (1895) \endchange \amendpage 2.559 line 5 of answer 6 (24.06.18) max \becomes sup \endchange \amendpage 2.560 lines 1--3 of answer 13 (24.06.18) (The ``max'' \dots notation.) For convenience \becomes For convenience \endchange \improvepage 2.561 in answer 16 (24.06.18) line $-5$: If we set $z\sqrt2=x_p$ and write \becomes If we now set\nl line $-1$: . \becomes , as reported for $x_p=z\sqrt2$ and $\nu>30$ in Table~1.
\endchange
\amendpage 2.572 line 3 (24.06.18)
$m\ll N$ \becomes $m$ is a lot smaller than $N$
\endchange
\improvepage 2.586 line 9 of answer 10 (24.06.30)
$x_j$ is the root of the equation \becomes $x_j$ satisfies
\endchange
\bugonpage 2.609 lines 4 and 5 of answer 30 (25.07.03)
2ドルb_jb_k$ \becomes 2ドル@\vert b_jb_k\vert$
\endchange
\amendpage 2.610 in answer 32 (24.07.22)
A more \becomes See exercise 1.2.6--11 and OEIS sequence A129508. A more
\endchange
\amendpage 2.628 line 9 of answer 39 (24.09.30)
913.] \becomes
913. The stated identity (and much more)
was derived by Euler in 1779,
but forgotten for hundreds of years! See M.~Ojanguren,
{\sl Elemente der Math.\ \bf77} (2022), 94--100.]\tighten\par\breathe
\endchange
\bugonpage 2.629 bottom line (24.07.29)
insert the factor $(-1)^r$ before `$m+1\choose r$'
\endchange
\bugonpage 2.645 in the Morris reference, beginning of answer 34 (22.11.10)
73--145 \becomes 73--143
\endchange
\improvepage 2.650 line 23 (24.07.29)
then \becomes where $_2F_0$ denotes a hypergeometric function, then
\endchange
\bugonpage 2.664 line 1 (24.07.29)
 55182 \becomes 54792, 55182
\endchange
\amendpage 2.667 in the last paragraph of answer 41 (22.12.29)
Further refinements \dots\ 768. \becomes
Further refinements due to numerous other researchers have made it possible to
go much higher; the current record, due to David Baugh and Kim Walisch
in 2022, is $\pi(10^{29})=1520698109714272166094258063$.
(See OEIS sequence A006880.)
\endchange
\amendpage 2.667 at the end of answer 41 (22.12.30)
large.] \becomes
large. See also the elementary method of D.~Hirsch, I.~Kessler, and
U.~Mendlovic, \arXiv:2212.09857 [math.NT] (2022), 60~pages.]
\endchange
\bugonpage 2.673 bottom line (22.12.08)
$\pm\ell_k^{n_{k-1}}\!@/\prod_{1<j<k}\ell_{j}^{\delta_{j-1}(\delta_j-1)}$. \becomes $\pm\prod_{j=2}^k \ell_j^{,円e_j},ドル where $e_j=\delta_{j-1}\prod_{i=j}^{k-1}(1-\delta_i),ドル $\delta_i=\deg(u_i)-\deg(u_{i+1})$. \endchange \bugonpage 2.677 line $-2$ of answer 22 (24.08.17) coefficients $y_j$ \becomes coefficients $u_j$ \endchange \improvepage 2.679 in last paragraph of answer 2 (24.08.17) line $-3$: $v(x)@w(x)^r$ \becomes $v(x)^rw(x)$\nl lines $-2$ and $-1$: $v(x)$ \becomes $w(x)$ \endchange \improvepage 2.686 line 1 of answer 26 (24.08.17) $D\gets(0\ldots01),ドル representing \becomes $D \gets 0 \ldots 0 1,ドル the $n$-bit vector that represents \endchange \bugonpage 2.693 line 1 of answer 15 (24.08.17) Diplomarbeit in Mathematics \becomes Diplomarbeit in Mathematik \endchange \amendpage 2.697 in answer 39 (22.08.14) line 8: 21.] \becomes 21.] In this argument $\{n_1,n_2,\ldots,n_m\}$ can be any {\it multiset}.\nl line 16: 2--3 \becomes 2--4 \endchange \amendpage 2.698 line 2 (22.02.22) for any $(r+1)\times(s+1)$ \becomes for any nonzero $(r+1)\times(s+1)$ \endchange \amendpage 2.698 bottom line of answer 41 (22.02.22) for $\{n_1,\ldots,n_m\}$ for odd numbers $\{n_1,\ldots,n_m\}$ \endchange \amendpage 2.698 bottom line of answer 42 (24.09.24) . \becomes . His website \.{https://www.additionchains.com/} provides extensive further details. \endchange \amendpage 2.700 near the end of answer 12 (22.11.10) \arXiv:1211.6320 [cs.CC] (2013), 10~pages. \becomes {\sl Linear Algebra and Its Applications\/ \bf 438} (2013), 4500--4509. \endchange \amendpage 7.715 the end of answer 62 (23.11.10) \def\\{\mathop{\hbox{rank}}}% 235. \becomes 235. For a modern treatment, including proofs that $\underline{\\}(T(2,2,3))=10$ and $\underline{\\}(T(2,3,3))=14,ドル see A.Conner, A.~Harper, and J.~M. Landsberg, {\sl Forum of Mathematics, Pi\/ \bf11} (2023), e17:1--30. \endchange \amendpage 7.715 last line of answer 64 (23.11.10) The border rank of $T(2,2,2)$ remains unknown.] \becomes The paper cited in answer 62 proves that $\underline{\\}(T(2,2,2))=7$ and $\underline{\\}(T(3,3,3))\ge17$.] \endchange \amendpage 7.725 new answer (24.04.01) \ans29. (a) The result is clear when $p,ドル $q,ドル $r,ドル and $s$ are constant. So it suffices to consider cases where $p(0)=q(0)=r(0)=s(0)=1$.\par Suppose $p(w)=1+p_1w+p_2w^2+\cdots$ and $q(z)=r(w)=s(z)=1$. Then $t_{m,n}=\break a_{m,n}+p_1a_{m-1,n}+\cdots+p_ma_{0,n}$. So every row of~$T_n$ is obtained from $A_n$ by adding multiples of previous rows (working bottom to top), leaving the determinant unchanged.\tighten\par Similarly, suppose $r(w)=1+r_1w+r_2w^2+\cdots$ and $p(w)=q(z)=s(z)=1$. Then we have, for example, $t_{5,n}=a_{5,n}+4r_1a_{4,n}+(3r_2+3r_1^2)@a_{3,n}+(2r_3+2r_2r_1)@a_{2,n} +r_4a_{1,n}$; in general every row is again a linear combination of its predecessors.\par (b) Setting $p(w)=1/(1+w),ドル $q(z)=1/(1+z),ドル $r(w)={1\over w}\ln(1+w),ドル $s(z)={1\over z}\ln(1+z)$ gives $T(w,z)=1/(1-wz)$. Hence $T_n$ is the identity matrix, and $\det A_n=1$.\par [See {\sl AMM\/ \bf86} (1979), 64--65; Ira~M.~Gessel and Guoce~Xin, {\sl Electronic Journal of Combinatorics\/ \bf13} (2006), \#R53, 1--48, \S4.] \endchange \amendpage 2.734 new entry {(after $x\approx y$)} (24.11.05) \tenpoint round($x$) $\big\vert$ nearest representable approximation to $x$ \ \ $\big\vert$ 4.2.2, 4.5.1 \endchange \expandafter\ifx\csname indexeject\endcsname\relax\else\vfill\eject\fi \amendpage 2.737 and following (22.02.01) Miscellaneous changes to the existing index of Volume~2 are collected here, including corrections and amendments to the old entries as well as new entries that are occasioned by the new material. Thus, the lines of the full index that have changed serve also as an index to the present document. However, when a correction or amendment has caused an old index entry to be deleted, the deletion is usually not indicated. \input exotic \begindoublecolumns \indexformat \gdef\Uni1.08:{\bitmap24:1.08:} \hangindent 2em al-Samaw'al (= as-Samaw'al), \indexbreak ibn Ya\d{h}y\=a ibn Yah\=uda al-Maghrib\ii\ \indexbreak ({\arab\Al/{\bigarab\Aham/}$\?\?$\Afw/\Amm/\Ams/\Ail/\Aa/ % see Hosny email \Afy/\Aib/\Afr/\Amgh/\Amm/\Ail/\Aa/ \Aa/\Ad/\Afw/\Amh/\Aiy/ \Afn/\Aib/ \Afam/\Amy/\Amhh/\Aiy/ \Afn/\Aib/\Aa/}, modern~form {\arab\Al/\Aha/\Afw/\Amm/\Ams/\Ail/\Aa/}), 198. % 47th \=Aryabha\d{t}a I ({\dn aA\hbox{y\kern-0.3em\hbox{\char13B}}V}), 198, 343. % 48th Baugh, David Dean, 667. % 48th Borel, F\'elix \smash{\'E}douard Justin \smash{\'E}mile, 177. % 49th Brent, Richard Peirce, 8, 28, 40, 130, 135, 139, 141, 187, 241, 279, 280, 313, 348, 352--353, 355, 356, 382, 386, 403, 501, 529--534, 539--540, 556, 590, 600, 643, 644, 646, 657, 658, 695, 719--721. % 51st Charles XII of Sweden (Karl XII), 200--201. % 47th Chebyshev (= Tschebyscheff), Pafnutii Lvovich ({\rus Chebyshevp2, Pafnut{\char12}i0 Lp1vovichp2} = {\rus Chebyshe0v, Pafnutii0 Lp1vovich}), inequality, 183, 669. % 50th Conner, Austin, 715. % 48th de La Vall\'ee Poussin, Charles Jean \'Etienne Gustave Nicolas, 381. % 50th Duality principle, 481, 485, 507, 535, 697, 718. % 47th Durant, Luke David, 409. % 50th Elkies, Noam David ({\heb\Hss/\Hqq/\Hll/\Haa/ \Hdd/\Haa/$\?$\Hdd/ \Hfm/\Hai/\Hnn/}), xi. % 50th Euler, Leonhard ({\rus Ei0lerp2, Leonardp2} = {\rus E1i0ler, Leonard}), xi, 357, 375, 377, 392, 407, 526, 628, 649--651, 653, 655. % 49th Fraenkel, Aviezri Siegmund\indexbreak ({\heb\Hll/\Hqq/\Hnn/\Hrr/\Hpp/ \Hdd/\Hnn/\Hvv/\Hmm/\Hgg/\Hyy/\Hss/ \Hyy/\Hrr/\Hzz/\Hai/\Hyy/\Hbb/\Haa/}), 290, 291, 292, 630. % 49th Gessel, Ira Martin, 537, 723. % 49th Gourdon, Xavier Richard, 355, 449. % 48th Harper, Alicia Deen, 715. % 48th Hexadecimal digits, 195, 201, 210. % 47th Hexadecimal number system, 200--202, 204, 210, 324, 639. % 47th Hill, Thomas Wright, 201. % 47th Hirsch, Dean ({\heb\Hsh/\Hrr/\Hyy/\Hhh/ \Hfn/\Hyy/\Hdd/}), 667. % 48th Hypergeometric functions, 650. % 49th Independence of random numbers, 2, 43, 46, 55, 59, 66, 95, 240, 560, 562. % 49th Indian mathematics, 197--198, 208, 209, 281, 287, 343, 387, 461, 648. % 48th Internet, iv, x, 698. % 49th \.{JAE} (jump if rA even), 339, 481. % 50th \.{JAO} (jump if rA odd), 339, 612. % 50th \.{JXE} (jump if rX even), 339. % 50th \.{JXO} (jump if rX odd), 219, 339. % 50th Kessler, Ido July ({\heb\Hrr/\Hll/\Hss/\Hqq/ \Hyy/\Hll/\Hvv/\Hyy/ \Hvv/\Hdd/\Hyy/\Hai/}), 667. % 48th Knuth, Donald Ervin (\GC75:75:-3:62% G2463 <0000001f0000000000000000001ff8000000000000000007ff000000000000000003ff80% 00000000000000007f8000000000000000007f8000000000000000003f8000003c000000% 00003f8000007e00000000001f800000ff00000000000f800001ff00ffffffffffffffff% ff80ffffffffffffffffffc07fffffffffffffffffe03800000000000000000000000000% 00000000000000000000000070000000000078000000f800000000007f000001fc000000% 00007ffffffffe00000000007fffffffff00000000003ffffffffe00000000003f000000% fc00000000003f000000fc00000000003f000000fc00000000003f000000fc0000000000% 3f000000fc00000000003f000000fc00000000003f000000fc00000000003f000000fc00% 000000003ffffffffc00000000003ffffffffc00000000003ffffffffc00000000003f00% 0000fc00000000003f000000fc00000000003f000000fc00000001c03e0000000003c000% 01e03e0000000007e00001f800000000000ff00001fffffffffffffff00001ffffffffff% fffff80001f8000000000007f80001f8000000000007e00001f8000000000007e00001f8% 000000000007e00001f801c000078007e00001f801e00007e007e00001f801f8000ff007% e00001f801fffffff007e00001f801fffffff007e00001f801ffffffc007e00001f801f8% 000fc007e00001f801f8000fc007e00001f801f8000fc007e00001f801f8000fc007e000% 01f801f8000fc007e00001f801f8000fc007e00001f801f8000fc007e00001f801f8000f% c007e00001f801f8000fc007e00001f801ffffffc007e00001f801ffffffc007e00001f8% 01ffffffc007e00001f801f8000fc007e00001f801f0000fc007e00001f801f0000f0007% e00001f801f000000007e00001f8000000000007e00001f800000000ffcfe00001f80000% 00007fffc00001f8000000000fffc00001f80000000000ffc00001f800000000007f8000% 01f800000000003f800001f000000000001f000001f00000000000100000>%% Unicode char "9ad8
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), ii, iv, vii, 2, 4, 30, 89, 138, 145, 159, 189, 196, 205, 226, 242, 316, 335, 373, 378, 384, 435, 477, 491, 584, 595, 599, 606, 636, 659, 661, 686, 694, 697, 722, 741, 764. % 47th
Krishnamurthy, Edayathumangalam Venkataraman ({\tm\103円\126円\175円\204円\127円\156円\316円\203円\141円\170円\207円\153円 \124円\210円\141円\170円\175円\kern.05em\205円\127円\203円\150円 \220円\315円\161円\176円\353円\155円\147円\227円}), 278, 279. % 48th
La Vall\'ee Poussin, Charles Jean \'Etienne Gustave Nicolas de, 381. % 50th
Landsberg, Joseph Montague, 715. % 48th
Lewis, Harry Roy, 200. % 47th
M\=adhava of Sangamagr\=ama ({\dn s\kern-.6em\raise.4em\hbox{\char21}gmg\llap{\char125}Amc\llap{\char3} mADv}), 198. % 48th
Mendlovich, Eliezer Uri\indexbreak ({\heb\Hfz/\Hyy/\Hbb/\Hvv/\Hll/\Hdd/\Hnn/\Hmm/ \Hyy/\Hrr/\Hvv/\Haa/ \Hrr/\Hzz/\Hai/\Hyy/\Hll/\Haa/}), 667. % 48th
\MIX\ computer, vi, 210. % 47th
(modulo $f(z)$ and $m$): 37. % 49th
Multisets, 170, 473, 483, 697. % 47th
N{\ii}lakantha Somayaji ({\dn nFlk{\char23}W somyAEj}), 198. % 48th
Number system, senidenary (= hexadecimal), 202. % 51st (was "-202", Aug22--Aug25)
OEIS\regtm: The On-Line Encyclopedia of Integer Sequences\regtm\ ({\tt oeis\period org}), 610, 667. % 49th
Ojanguren, Manuel Martin, 628. % 49th
Oliveira e Silva, Tom\'as Ant\'onio Mendes, 386. % 48th
Ostrowski, Alexander Markowich ({\rus Ostrovsp1kii0, Oleksandr Markovich}), 379, 494. % 50th
Pelik\'an (= Pelicanus), V\'aclav Josef, 200. % 49th
Random point, in a sphere, 135. % 51st
Reeken, Anthonius (= Ton) Johannes van, 232. % 49th
R\'enyi, Alfr\'ed, 209. % 47th
Sphere or ball, in $n$ dimensions, random point in, 135. % 51st
Square root, 122, 200, 213, 283, 374--375, 397--398, 483. % 49th
Strickland, Lloyd, 200. % 47th
Thurber, Edward Gerrish, II, 466, 470, 477, 478. % 48th
Vall\'ee Poussin, Charles Jean \'Etienne Gustave Nicolas de La, 381. % 50th
van Reeken, Anthonius (= Ton) Johannes, 232. % 49th
Wales, Francis Herbert, 194. % 49th
Walisch, Kim Georges, 667. % 48th
Wiedijk, Frederik [= Freek], 665. % 49th
Word size, 12--16, 34, 265, 276. % 49th
Xin, Guoce (\GC75:74:-3:60% G4833
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), 725. % 49th
Zlatopolski, Dmitry Mikhailovich ({\rus Zlatopolp1skii0, Dmitrii0 Mikhai0lovich}), 200. % 49th
Zuse, Konrad Ernst Otto, 202, 225, 227. % 49th
\vfill
\enddoublecolumns
\endchange
\bye
[The next printing will be the 52nd.]
not listed above:
page 108, line 8, :\ Academic
page 109, line -16 and line -15, :\ Springer
page 114, line -18, :\ Wiley
page 115, line 5, :\ Academic
page 135, line -1, :\ McGraw--Hill
that change affects page 136
page 145, line -4, :\ ACM
page 146, line 17, :\ Academic
page 177, line 22, :\ Wiley
page 189, near the middle, :\ Kluwer [and :\ Methuen but it changes nothing]
page 192, line -1, :\ J. B. 
page 196, near the middle, :\ Princeton , :\ P. Noordhoff
page 197, near the middle, :\ Ginn
page 199, near the top, "(Oxford, 1685)" "(Paris, 1963)" "(London, 1687)"
page 200, near the top, "(Paris, 1703)"
page 200, near the middle, :\ Freeman
page 201, near the top, :\ E. R. Dumont
page 201, near the top, "(London, 1860)"
page 201, near the middle, "(Paris, 1845)" , :\ Le Nonnant
page 202, near the middle, :\ Springer
page 225, line 12, :\ Princeton
page 225, line -11, :\ Springer
page 226, lines 5 and 13, :\ Macmillan , :\ Harvard
page 226, lines -16 and -13 and -10, :\ Wiley :\ Wiley :\ McGraw
page 240, near the middle, :\ Wiley
page 278, line -11, "(Paris, 1831)"
page 278, line -4, :\ McGraw
page 279, line -4, "(London, 1853)"
page 280, line -13 and line -10, :\ Academic
page 292, line 18, :\ McGraw
page 326, bottom line, "(Paris, 1798)"
page 332, line 20, :\ Academic
page 341, top line, "(Leipzig, 1913)"
page 345, line 14, :\ Springer
page 348, line -12, :\ Academic
page 382, lines 6 and 7, "(Berlin, 1963)"
page 407, line -3, "(Paris, 1924)"
page 422, line -3, "(Leipzig, 1801)"
page 433, line -5, :\ Ungar
page 461, line -10, "(London, 1879)"
page 494, line 17, :\ Academic
page 498, line 5, :\ Academic
page 503, line 18, "(Paris, 1877)"
page 505, lines -10 and -9, :\ Academic :\ Birkh\"auser
page 515, lines 5,6,7, :\ American , :\ Birkh\"auser , :\ Springer
page 518, line 7, :\ Ungar
page 539 in answer 6(b), "(Berlin, 1895)"
page 556, line -4, :\ ACM
page 578, last line of answer 4, "(Leipzig, 1801)"
page 584, line -9, :\ ACM
page 597, ans 31, "(Paris, 1939)"
page 622, line 6 of answer 19, "(Moscow, 1954)"
page 627, line 5 of answer 36, :\ Longmans
page 636, last line of answer 3, :\ Springer
page 648, line 4 and line 5, :\ Hutchinson , :\ Addison
page 649, end of answer 13, "(Berlin, 1768)"
page 653, line 7, "(Berlin, 1768)"
page 655, line 6, "(Berlin, 1767)"
page 676, line 1 of answer 19, "(Paris, 1914)"
page 678, line -6 of answer 26, "(Berlin, 1881)"
page 688, line -2 of answer 32(b), "(Leipzig, 1801)"
page 689, line -2, "(Berlin, 1896)"
page 697, line 8 of answer 38, :\ Birkh\"auser
page 699, in answer 11, :\ Birkh\"auser
page 719, last line of answer 4, :\ Academic
page 721, in answer 13, :\ Springer , :\ Birkh\"auser
ARTICLES "TO APPEAR" THAT ARE STILL PENDING:
(I last checked GIMPS on 17 July 2025)
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