COMMENT ⊗   VALID 00010 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	\input acphdr % This file contains material to be set out of normal sequence
C00003 00003	% Table 1 from Section 3.2.1.1	*
C00011 00004	% Table 3.3.4-1 (goes on top of two facing pages)	*
C00023 00005	% Table 1 from Section 4.3.3	*
C00026 00006	% Table 1 from Section 4.5.4	*
C00031 00007	% Table 1 from Section 4.6.1	*
C00036 00008	% Appendix A (Tables of numerical quantities)
C00053 00009	% Appendix B (Index to notations)	*
C00073 00010	\end % eject previous page and terminate
C00074 ENDMK
C⊗;
\input acphdr % This file contains material to be set out of normal sequence
\open0=v2spec.inx
\titlepage\setcount00
\null
\vfill
\tenpoint
\ctrline{Volume 2 of THE ART OF COMPUTER PROGRAMMING}
\ctrline{---appendices and special tables set out of normal sequence---}
\ctrline{$\copyright$ 1980 Addison--Wesley Publishing Company, Inc.}
\vfill
\runninglefthead{RANDOM NUMBERS}
% The page numbers may need to be changed
% Table 1 from Section 3.2.1.1	*
\runningrighthead{CHOICE OF MODULUS}
\section{3.2.1.1}
\eject % eject the previous page
\acpmark{\chd}{\csec}
\setcount0 13
\tablehead{Table 1}
\penalty1000
\vfill
\ctrline{\hbox expand 5pt{PRIME FACTORIZATIONS OF $w \pm 1$}}
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⊗\¬⊗⊗\¬\cr
7 \cdot 31 \cdot 151⊗\\⊗15⊗\\⊗3↑2 \cdot 11 \cdot 331\cr
3 \cdot 5 \cdot 17 \cdot 257⊗\\⊗16⊗\\⊗65537\cr
131071⊗\\⊗17⊗\\⊗3 \cdot 43691\cr
3↑3 \cdot 7 \cdot 19 \cdot 73⊗\\⊗18⊗\\⊗5 \cdot 13 \cdot 37 \cdot 109\cr
524287⊗\\⊗19⊗\\⊗3 \cdot 174763\cr
3 \cdot 5↑2 \cdot 11 \cdot 31 \cdot 41⊗\\⊗20⊗\\⊗17 \cdot 61681\cr
7↑2 \cdot 127 \cdot 337⊗\\⊗21⊗\\⊗3↑2 \cdot 43 \cdot 5419\cr
3 \cdot 23 \cdot 89 \cdot 683⊗\\⊗22⊗\\⊗5 \cdot 397 \cdot 2113\cr
47 \cdot 178481⊗\\⊗23⊗\\⊗3 \cdot 2796203\cr
3↑2 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \cdot 241⊗\\⊗24⊗\\⊗97 \cdot 257 \cdot 673\cr
31 \cdot 601 \cdot 1801⊗\\⊗25⊗\\⊗3 \cdot 11 \cdot 251 \cdot 4051\cr
3 \cdot 2731 \cdot 8191⊗\\⊗26⊗\\⊗5 \cdot 53 \cdot 157 \cdot 1613\cr
7 \cdot 73 \cdot 262657⊗\\⊗27⊗\\⊗3↑4 \cdot 19 \cdot 87211\cr
3 \cdot 5 \cdot 29 \cdot 43 \cdot 113 \cdot 127⊗\\⊗28⊗\\⊗17 \cdot 15790321\cr
233 \cdot 1103 \cdot 2089⊗\\⊗29⊗\\⊗3 \cdot 59 \cdot 3033169\cr
3↑2 \cdot 7 \cdot 11 \cdot 31 \cdot 151 \cdot 331⊗\\⊗30⊗\\⊗5↑2 \cdot
	13 \cdot 41 \cdot 61 \cdot 1321\cr
2147483647⊗\\⊗31⊗\\⊗3 \cdot 715827883\cr
3 \cdot 5 \cdot 17 \cdot 257 \cdot 65537⊗\\⊗32⊗\\⊗641 \cdot 6700417\cr
7 \cdot 23 \cdot 89 \cdot 599479⊗\\⊗33⊗\\⊗3↑2 \cdot 67 \cdot 683 \cdot 20857\cr
3 \cdot 43691 \cdot 131071⊗\\⊗34⊗\\⊗5 \cdot 137 \cdot 953 \cdot 26317\cr
31 \cdot 71 \cdot 127 \cdot 122921⊗\\⊗35⊗\\⊗3 \cdot 11 \cdot 43 \cdot
	281 \cdot 86171\cr
3↑3 \cdot 5 \cdot 7 \cdot 13 \cdot 19 \cdot 37 \cdot 73 \cdot
	109⊗\\⊗36⊗\\⊗17 \cdot 241 \cdot 433 \cdot 38737\cr
223 \cdot 616318177⊗\\⊗37⊗\\⊗3 \cdot 1777 \cdot 25781083\cr
3 \cdot 174763 \cdot 524287⊗\\⊗38⊗\\⊗5 \cdot 229 \cdot 457 \cdot 525313\cr
7 \cdot 79 \cdot 8191 \cdot 121369⊗\\⊗39⊗\\⊗3↑2 \cdot 2731 \cdot 22366891\cr
3 \cdot 5↑2 \cdot 11 \cdot 17 \cdot 31 \cdot 41 \cdot 61681⊗\\⊗40⊗\\⊗257
	\cdot 4278255361\cr
13367 \cdot 164511353⊗\\⊗41⊗\\⊗3 \cdot 83 \cdot 8831418697\cr
3↑2 \cdot 7↑2 \cdot 43 \cdot 127 \cdot 337 \cdot 5419⊗\\⊗42⊗\\⊗5 \cdot
	13 \cdot 29 \cdot 113 \cdot 1429 \cdot 14449\cr
431 \cdot 9719 \cdot 2099863⊗\\⊗43⊗\\⊗3 \cdot 2932031007403\cr
3 \cdot 5 \cdot 23 \cdot 89 \cdot 397 \cdot 683 \cdot 2113⊗\\⊗44⊗\\⊗17
	\cdot 353 \cdot 2931542417\cr
7 \cdot 31 \cdot 73 \cdot 151 \cdot 631 \cdot 23311⊗\\⊗45⊗\\⊗3↑3 \cdot
	11 \cdot 19 \cdot 331 \cdot 18837001\cr
3 \cdot 47 \cdot 178481 \cdot 2796203⊗\\⊗46⊗\\⊗5 \cdot 277 \cdot 1013
	\cdot 1657 \cdot 30269\cr
2351 \cdot 4513 \cdot 13264529⊗\\⊗47⊗\\⊗3 \cdot 283 \cdot 165768537521\cr
3↑2 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \cdot 97 \cdot 241 \cdot
	257 \cdot 673⊗\\⊗48⊗\\⊗193 \cdot 65537 \cdot 22253377\cr
179951 \cdot 3203431780337⊗\\⊗59⊗\\⊗3 \cdot 2833 \cdot 37171 \cdot 1824726041\cr
3↑2 \cdot 5↑2 \cdot 7 \cdot 11 \cdot 13 \cdot 31 \cdot 41 \cdot
	61 \cdot 151 \cdot 331 \cdot 1321⊗\\⊗60⊗\\⊗17 \cdot 241 \cdot 61681
	\cdot 4562284561\cr
7↑2 \cdot 73 \cdot 127 \cdot 337 \cdot 92737
	\cdot 649657⊗\\⊗63⊗\\⊗3↑3 \cdot 19 \cdot 43 \cdot 5419 \cdot 77158673929\cr
3 \cdot 5 \cdot 17 \cdot 257 \cdot 641 \cdot 65537 \cdot 6700417⊗\\⊗64⊗\\⊗274177
	\cdot 67280421310721\cr
⊗\¬⊗⊗\¬\cr
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3↑3 \cdot 7 \cdot 11 \cdot 13 \cdot 37⊗\\⊗\96⊗\\⊗101 \cdot 9901\cr
3↑2 \cdot 239 \cdot 4649⊗\\⊗\97⊗\\⊗11 \cdot 909091\cr
3↑2 \cdot 11 \cdot 73 \cdot 101 \cdot 137⊗\\⊗\98⊗\\⊗17 \cdot 5882353\cr
3↑4 \cdot 37 \cdot 333667⊗\\⊗\99⊗\\⊗7 \cdot 11 \cdot 13 \cdot 19 \cdot 52579\cr
3↑2 \cdot 11 \cdot 41 \cdot 271 \cdot 9091⊗\\⊗10⊗\\⊗101 \cdot 3541
	\cdot 27961\cr
3↑2 \cdot 21649 \cdot 513239⊗\\⊗11⊗\\⊗11↑2 \cdot 23 \cdot 4093 \cdot 8779\cr
3↑3 \cdot 7 \cdot 11 \cdot 13 \cdot 37 \cdot 101 \cdot 9901⊗\\⊗12⊗\\⊗73
	\cdot 137 \cdot 99990001\cr
3↑2 \cdot 11 \cdot 17 \cdot 73 \cdot 101 \cdot
	137 \cdot 5882353⊗\\⊗16⊗\\⊗353 \cdot 449 \cdot 641 \cdot 1409 \cdot 69857\cr
⊗\¬⊗⊗\¬\cr
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% Table 3.3.4-1 (goes on top of two facing pages)	*
% This table is accounted for by two topinserts of height 360pt
\runningrighthead{THE SPECTRAL TEST}
\section{3.3.4}
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2⊗2↑7 + 1⊗2↑{35}⊗16642 ⊗16642 ⊗16642 ⊗15602 ⊗252 \cr
3⊗2↑{18} + 1⊗2↑{35}⊗34359738368 ⊗6 ⊗4 ⊗4 ⊗4 \cr
4⊗3141592653 ⊗2↑{35}⊗2997222016 ⊗1026050 ⊗27822 ⊗1118 ⊗1118 \cr
5⊗137 ⊗256⊗274 ⊗30 ⊗14 ⊗6 ⊗4 \cr
6⊗3141592621 ⊗10↑{10}⊗4577114792 ⊗1034718 ⊗62454 ⊗1776 ⊗542 \cr
7⊗3141592221 ⊗10↑{10}⊗4293881050 ⊗276266 ⊗97450 ⊗3366 ⊗2382 \cr
8⊗4219755981 ⊗10↑{10}⊗10721093248 ⊗2595578 ⊗49362 ⊗5868 ⊗820 \cr
9⊗4160984121 ⊗10↑{10}⊗9183801602 ⊗4615650 ⊗16686 ⊗6840 ⊗1344 \cr
10⊗3141592221 ⊗2↑{35}⊗13539813818 ⊗5795090 ⊗88134 ⊗12716 ⊗2938 \cr
11⊗2718281829 ⊗2↑{35}⊗22939188896 ⊗2723830 ⊗146116 ⊗10782 ⊗2914 \cr
12⊗5↑{13}⊗2↑{35}⊗33161885770 ⊗2925242 ⊗113374 ⊗13070 ⊗2256 \cr
13⊗5↑{15}⊗2↑{35}⊗22078865098 ⊗10274746 ⊗167558 ⊗5844 ⊗2592 \cr
14⊗2↑{23} + 2↑{12} + 5⊗2↑{35}⊗167510120 ⊗8052254 ⊗21476 ⊗16802
⊗1630 \cr
15⊗2↑{23} + 2↑{13} + 5⊗2↑{35}⊗168231328 ⊗5335322 ⊗21476 ⊗2008
⊗1134 \cr
16⊗2↑{23} + 2↑{14} + 5⊗2↑{35}⊗12256151168 ⊗5733878 ⊗21476 ⊗13316
⊗2032 \cr
17⊗2↑{22} + 2↑{13} + 5⊗2↑{35}⊗8201443840 ⊗1830230 ⊗21476 ⊗7786
⊗3080 \cr
18⊗2↑{24} + 2↑{13} + 5⊗2↑{35}⊗8364058⊗8364058⊗21476 ⊗16712 ⊗1496 \cr
19⊗19935388837 ⊗2↑{35}⊗32300850938 ⊗705518 ⊗22270 ⊗9558 ⊗2660 \cr
20⊗1175245817 ⊗2↑{35}⊗36436418002 ⊗7362242 ⊗95306 ⊗3006 ⊗2860 \cr
21⊗17059465 ⊗2↑{35}⊗39341117000 ⊗9476606 ⊗202796 ⊗18758 ⊗2382 \cr
22⊗2↑{16} + 3⊗2↑{29}⊗536805386 ⊗118 ⊗116 ⊗116 ⊗116 \cr
23⊗1812433253 ⊗2↑{32}⊗4326934538 ⊗1462856 ⊗15082 ⊗4866 ⊗906 \cr
24⊗1566083941 ⊗2↑{32}⊗4659748970 ⊗2079590 ⊗44902 ⊗4652 ⊗662 \cr
25⊗69069 ⊗2↑{32}⊗4243209856 ⊗2072544 ⊗52804 ⊗6990 ⊗242 \cr
26⊗1664525 ⊗2↑{32}⊗4938916874 ⊗2322494 ⊗63712 ⊗4092 ⊗1038 \cr
27⊗314159269 ⊗2↑{31} - 1⊗1432232969 ⊗899290 ⊗36985 ⊗3427 ⊗1144 \cr
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29⊗31167285 ⊗2↑{48}⊗3.2 \times 10↑{14}⊗4111841446 ⊗17341510
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30⊗\hbox{see the text}⊗2↑{64} ⊗8.8\times10↑{18}⊗6.4\times10↑{12}⊗
4.1\times10↑9⊗45662836⊗1846368\cr
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4.5⊗4.5⊗4.5⊗4.5⊗4.4⊗2\epsilon↑5⊗5\epsilon↑4⊗0.01⊗0.34⊗4.62⊗1\cr
7.0⊗7.0⊗7.0⊗7.0⊗4.0⊗2\epsilon↑6⊗3\epsilon↑4⊗0.04⊗4.66⊗2\epsilon
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17.5⊗1.3⊗1.0⊗1.0⊗1.0⊗3.14⊗2\epsilon↑9⊗2\epsilon↑9⊗5\epsilon
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15.7⊗10.0⊗7.4⊗5.0⊗5.0⊗0.27⊗0.13⊗0.11⊗0.01⊗0.21⊗4\cr
4.0⊗2.5⊗1.9⊗1.3⊗1.0⊗3.36⊗2.69⊗3.78⊗1.81⊗1.29⊗5\cr
16.0⊗10.0⊗8.0⊗5.4⊗4.5⊗1.44⊗0.44⊗1.92⊗0.07⊗0.08⊗6\cr
16.0⊗9.0⊗8.3⊗5.9⊗5.6⊗1.35⊗0.06⊗4.69⊗0.35⊗6.98⊗7\cr
16.7⊗10.7⊗7.8⊗6.3⊗4.8⊗3.39⊗1.75⊗1.20⊗1.39⊗0.28⊗8\cr
16.5⊗11.1⊗7.0⊗6.4⊗5.2⊗2.89⊗4.15⊗0.14⊗2.04⊗1.25⊗9\cr
16.8⊗11.2⊗8.2⊗6.8⊗5.8⊗1.24⊗1.70⊗1.12⊗2.79⊗3.81⊗10\cr
17.2⊗10.7⊗8.6⊗6.7⊗5.8⊗2.10⊗0.55⊗3.15⊗1.85⊗3.72⊗11\cr
17.5⊗10.7⊗8.4⊗6.8⊗5.6⊗3.03⊗0.61⊗1.85⊗2.99⊗1.73⊗12\cr
17.2⊗11.6⊗8.7⊗6.3⊗5.7⊗2.02⊗4.02⊗4.03⊗0.40⊗2.62⊗13\cr
13.7⊗11.5⊗7.2⊗7.0⊗5.3⊗0.02⊗2.79⊗0.07⊗5.61⊗0.65⊗14\cr
13.7⊗11.2⊗7.2⊗5.5⊗5.1⊗0.02⊗1.50⊗0.07⊗0.03⊗0.22⊗15\cr
16.8⊗11.2⊗7.2⊗6.9⊗5.5⊗1.12⊗1.67⊗0.07⊗3.13⊗1.26⊗16\cr
16.5⊗10.4⊗7.2⊗6.5⊗5.8⊗0.75⊗0.30⊗0.07⊗0.82⊗4.39⊗17\cr
11.5⊗11.5⊗7.2⊗7.0⊗5.3⊗8\epsilon↑4⊗2.95⊗0.07⊗5.53⊗0.50⊗18\cr
17.5⊗ 9.7⊗7.2⊗6.6⊗5.7⊗2.95⊗0.07⊗0.07⊗1.37⊗2.83⊗19\cr
17.5⊗11.4⊗8.3⊗5.8⊗5.7⊗3.33⊗2.44⊗1.30⊗0.08⊗3.52⊗20\cr
17.6⊗11.6⊗8.8⊗7.1⊗5.6⊗3.60⊗3.56⊗5.91⊗7.38⊗2.03⊗21\cr
14.5⊗3.4⊗3.4⊗3.4⊗3.4⊗3.14⊗\epsilon↑5⊗\epsilon↑4⊗\epsilon
↑3⊗0.02⊗22\cr
16.0⊗10.2⊗6.9⊗6.1⊗4.9⊗3.16⊗1.73⊗0.26⊗2.02⊗0.89⊗23\cr
16.1⊗10.5⊗7.7⊗6.1⊗4.7⊗3.41⊗2.92⊗2.32⊗1.81⊗0.35⊗24\cr
16.0⊗10.5⊗7.8⊗6.4⊗4.0⊗3.10⊗2.91⊗3.20⊗5.01⊗0.02⊗25\cr
16.1⊗10.6⊗8.0⊗6.0⊗5.0⊗3.61⊗3.45⊗4.66⊗1.31⊗1.35⊗26\cr
15.2⊗9.9⊗7.6⊗5.9⊗5.1⊗2.10⊗1.66⊗3.14⊗1.69⊗3.60⊗27\cr
31.0⊗20.2⊗15.6⊗11.8⊗9.8⊗3.14⊗1.49⊗0.44⊗0.69⊗0.66⊗28\cr
24.1⊗16.0⊗12.0⊗9.1⊗7.9⊗3.60⊗3.92⊗5.27⊗0.97⊗3.82⊗29\cr
31.5⊗21.3⊗16.0⊗12.7⊗10.4⊗1.50⊗3.68⊗4.52⊗4.02⊗1.76⊗30\cr
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\hbox{\vbox to 9.778pt{}\hbox to 122pt{\hfill upper bounds from (40): }\!
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\hbox to 28pt{\hfill\hskip-4pt14.89\hfill}\!
\hbox to 28pt{\hfill\hskip-4pt23.87\hfill}}
\vfill % end right half of table
% Table 1 from Section 4.3.3	*
\runninglefthead{ARITHMETIC}
\runningrighthead{HOW FAST CAN WE MULTIPLY\:a?}
\section{4.3.3}
\eject % eject the previous page
\acpmark{\chd}{\csec}
\setcount0 298
\vskip 6pt
\tablehead{Table 1}
\penalty1000
\vfill
\ctrline{\hbox expand 8pt{MULTIPLICATION IN A LINEAR ITERATIVE ARRAY}}
\ninepoint
\penalty1000
\vfill
\lineskip 0pt plus 100pt
\penalty1000
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⊗\hfill#\hfill⊗\hfill#\hfill⊗\hfill#\hfill⊗\hfill#\hfill\tabskip0pt\cr
Time⊗Input⊗Module $M↓1$⊗Module $M↓2$⊗Module $M↓3$\cr
\noalign{\vskip 3pt\hrule\vskip 3pt}
⊗\\{u↓j}{v↓j}{q↓j}⊗\!
\[c{x↓0}{y↓0}{x↓1}{y↓1}xy{z↓2}{z↓1}{z↓0}]⊗\!
\[c{x↓0}{y↓0}{x↓1}{y↓1}xy{z↓2}{z↓1}{z↓0}]⊗\!
\[c{x↓0}{y↓0}{x↓1}{y↓1}xy{z↓2}{z↓1}{z↓0}]\cr
\noalign{\vskip 3pt\hrule\vskip 3pt}
\90⊗\\111⊗\[0000000000]⊗\[0000000000]⊗\[0000000000]\cr
\91⊗\\111⊗\[1110000010]⊗\[0000000000]⊗\[0000000000]\cr
\92⊗\\110⊗\[2111100100]⊗\[0000000000]⊗\[0000000000]\cr
\93⊗\\001⊗\[3111111011]⊗\[0000000001]⊗\[0000000000]\cr
\94⊗\\110⊗\[3111100101]⊗\[1110000001]⊗\[0000000000]\cr
\95⊗\\000⊗\[3111111011]⊗\[2110000001]⊗\[0000000000]\cr
\96⊗\\000⊗\[3111100100]⊗\[3110011010]⊗\[0000000000]\cr
\97⊗\\000⊗\[3111100000]⊗\[3110000010]⊗\[1110000001]\cr
\98⊗\\000⊗\[3111100000]⊗\[3110000010]⊗\[2110000000]\cr
\99⊗\\000⊗\[3111100000]⊗\[3110000001]⊗\[3110000000]\cr
10⊗\\000⊗\[3111100001]⊗\[3110000000]⊗\[3110000000]\cr
11⊗\\000⊗\[3111100000]⊗\[3110000000]⊗\[3110000000]\cr}
\lineskip0pt
% Table 1 from Section 4.5.4	*
\runninglefthead{ARITHMETIC}
\runningrighthead{FACTORING INTO PRIMES}
\section{4.5.4}
\eject % eject the previous page
\acpmark{\chd}{\csec}
\setcount0 390
\tablehead{Table 1}
\vskip 3pt
\ctrline{\hbox expand 3pt{USEFUL PRIME NUMBERS}}
\ninepoint
\vskip 7pt
\hrule
\vskip 5pt
\baselineskip 10pt plus 1pt
\halign to size{$#\hfill$\tabskip 0pt plus 10pt
⊗\hfill#⊗\hfill#⊗\hfill#⊗\hfill#⊗\hfill#
⊗\hfill#⊗\hfill#⊗\hfill#⊗\hfill#⊗\hfill#\tabskip0pt\cr
\hfill N⊗$a↓1$⊗$a↓2$⊗$a↓3$⊗$a↓4$⊗$a↓5$⊗$a↓6$⊗$a↓7$⊗$a↓8$⊗$a↓9$⊗$a↓{10}$\cr
\noalign{\vskip 3pt}
2↑{15}⊗19⊗49⊗51⊗55⊗61⊗75⊗81⊗115⊗121⊗135\cr
2↑{16}⊗15⊗17⊗39⊗57⊗87⊗89⊗99⊗113⊗117⊗123\cr
2↑{17}⊗1⊗9⊗13⊗31⊗49⊗61⊗63⊗85⊗91⊗99\cr
2↑{18}⊗5⊗11⊗17⊗23⊗33⊗35⊗41⊗65⊗75⊗93\cr
2↑{19}⊗1⊗19⊗27⊗31⊗45⊗57⊗67⊗69⊗85⊗87\cr
2↑{20}⊗3⊗5⊗17⊗27⊗59⊗69⊗129⊗143⊗153⊗185\cr
2↑{21}⊗9⊗19⊗21⊗55⊗61⊗69⊗105⊗111⊗121⊗129\cr
2↑{22}⊗3⊗17⊗27⊗33⊗57⊗87⊗105⊗113⊗117⊗123\cr
2↑{23}⊗15⊗21⊗27⊗37⊗61⊗69⊗135⊗147⊗157⊗159\cr
2↑{24}⊗3⊗17⊗33⊗63⊗75⊗77⊗89⊗95⊗117⊗167\cr
2↑{25}⊗39⊗49⊗61⊗85⊗91⊗115⊗141⊗159⊗165⊗183\cr
2↑{26}⊗5⊗27⊗45⊗87⊗101⊗107⊗111⊗117⊗125⊗135\cr
2↑{27}⊗39⊗79⊗111⊗115⊗135⊗187⊗199⊗219⊗231⊗235\cr
2↑{28}⊗57⊗89⊗95⊗119⊗125⊗143⊗165⊗183⊗213⊗273\cr
2↑{29}⊗3⊗33⊗43⊗63⊗73⊗75⊗93⊗99⊗121⊗133\cr
2↑{30}⊗35⊗41⊗83⊗101⊗105⊗107⊗135⊗153⊗161⊗173\cr
2↑{31}⊗1⊗19⊗61⊗69⊗85⊗99⊗105⊗151⊗159⊗171\cr
2↑{32}⊗5⊗17⊗65⊗99⊗107⊗135⊗153⊗185⊗209⊗267\cr
2↑{33}⊗9⊗25⊗49⊗79⊗105⊗285⊗301⊗303⊗321⊗355\cr
2↑{34}⊗41⊗77⊗113⊗131⊗143⊗165⊗185⊗207⊗227⊗281\cr
2↑{35}⊗31⊗49⊗61⊗69⊗79⊗121⊗141⊗247⊗309⊗325\cr
2↑{36}⊗5⊗17⊗23⊗65⊗117⊗137⊗159⊗173⊗189⊗233\cr
2↑{37}⊗25⊗31⊗45⊗69⊗123⊗141⊗199⊗201⊗351⊗375\cr
2↑{38}⊗45⊗87⊗107⊗131⊗153⊗185⊗191⊗227⊗231⊗257\cr
2↑{39}⊗7⊗19⊗67⊗91⊗135⊗165⊗219⊗231⊗241⊗301\cr
2↑{40}⊗87⊗167⊗195⊗203⊗213⊗285⊗293⊗299⊗389⊗437\cr
2↑{41}⊗21⊗31⊗55⊗63⊗73⊗75⊗91⊗111⊗133⊗139\cr
2↑{42}⊗11⊗17⊗33⊗53⊗65⊗143⊗161⊗165⊗215⊗227\cr
2↑{43}⊗57⊗67⊗117⊗175⊗255⊗267⊗291⊗309⊗319⊗369\cr
2↑{44}⊗17⊗117⊗119⊗129⊗143⊗149⊗287⊗327⊗359⊗377\cr
2↑{45}⊗55⊗69⊗81⊗93⊗121⊗133⊗139⊗159⊗193⊗229\cr
2↑{46}⊗21⊗57⊗63⊗77⊗167⊗197⊗237⊗287⊗305⊗311\cr
2↑{47}⊗115⊗127⊗147⊗279⊗297⊗339⊗435⊗541⊗619⊗649\cr
2↑{48}⊗59⊗65⊗89⊗93⊗147⊗165⊗189⊗233⊗243⊗257\cr
2↑{59}⊗55⊗99⊗225⊗427⊗517⊗607⊗649⊗687⊗861⊗871\cr
2↑{60}⊗93⊗107⊗173⊗179⊗257⊗279⊗369⊗395⊗399⊗453\cr
2↑{63}⊗25⊗165⊗259⊗301⊗375⊗387⊗391⊗409⊗457⊗471\cr
2↑{64}⊗59⊗83⊗95⊗179⊗189⊗257⊗279⊗323⊗353⊗363\cr
\noalign{\vskip3pt}
10↑6⊗17⊗21⊗39⊗41⊗47⊗69⊗83⊗93⊗117⊗137\cr
10↑7⊗9⊗27⊗29⊗57⊗63⊗69⊗71⊗93⊗99⊗111\cr
10↑8⊗11⊗29⊗41⊗59⊗69⊗153⊗161⊗173⊗179⊗213\cr
10↑9⊗63⊗71⊗107⊗117⊗203⊗239⊗243⊗249⊗261⊗267\cr
10↑{10}⊗33⊗57⊗71⊗119⊗149⊗167⊗183⊗213⊗219⊗231\cr
10↑{11}⊗23⊗53⊗57⊗93⊗129⊗149⊗167⊗171⊗179⊗231\cr
10↑{12}⊗11⊗39⊗41⊗63⊗101⊗123⊗137⊗143⊗153⊗233\cr
10↑{16}⊗63⊗83⊗113⊗149⊗183⊗191⊗329⊗357⊗359⊗369\cr}
\vskip 5pt
\hrule
\vskip 5pt
\ctrline{The ten largest primes less than $N$ are $N-a↓1$, $\ldotss$, $N-a↓{10}$.}
% Table 1 from Section 4.6.1	*
\runninglefthead{ARITHMETIC}
\runningrighthead{DIVISION OF POLYNOMIALS}
\section{4.6.1}
\eject % eject the previous page
\acpmark{\chd}{\csec}
\setcount0 411
\tablehead{Table 1}
\vskip3pt
\ctrline{\hbox expand 5pt{COEFFICIENTS IN ALGORITHM C}}
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\+{name}⊗\\⊗⊗⊗⊗⊗⊗⊗\hbox{ R\spose{ow}}⊗⊗⊗⊗⊗⊗⊗⊗\\⊗\+{by}⊗\\⊗\hbox{\hskip-10pt
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⊗\¬⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗\¬⊗⊗\¬\cr
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⊗\¬⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗\¬⊗⊗\¬\cr
A↓5⊗\\⊗a↓8⊗a↓7⊗a↓6⊗a↓5⊗a↓4⊗a↓3⊗a↓2⊗a↓1⊗a↓0⊗0⊗0⊗0⊗0⊗0⊗\\⊗b↓6↑3⊗\\⊗C↓5\cr
A↓4⊗\\⊗0⊗a↓8⊗a↓7⊗a↓6⊗a↓5⊗a↓4⊗a↓3⊗a↓2⊗a↓1⊗a↓0⊗0⊗0⊗0⊗0⊗\\⊗b↓6↑3⊗\\⊗C↓4\cr
A↓3⊗\\⊗0⊗0⊗a↓8⊗a↓7⊗a↓6⊗a↓5⊗a↓4⊗a↓3⊗a↓2⊗a↓1⊗a↓0⊗0⊗0⊗0⊗\\⊗b↓6↑3⊗\\⊗C↓3\cr
A↓2⊗\\⊗0⊗0⊗0⊗a↓8⊗a↓7⊗a↓6⊗a↓5⊗a↓4⊗a↓3⊗a↓2⊗a↓1⊗a↓0⊗0⊗0⊗\\⊗b↓6↑3⊗\\⊗C↓2\cr
A↓1⊗\\⊗0⊗0⊗0⊗0⊗a↓8⊗a↓7⊗a↓6⊗a↓5⊗a↓4⊗a↓3⊗a↓2⊗a↓1⊗a↓0⊗0⊗\\⊗b↓6↑3⊗\\⊗C↓1\cr
A↓0⊗\\⊗0⊗0⊗0⊗0⊗0⊗a↓8⊗a↓7⊗a↓6⊗a↓5⊗a↓4⊗a↓3⊗a↓2⊗a↓1⊗a↓0⊗\\⊗b↓6↑3⊗\\⊗C↓0\cr
B↓7⊗\\⊗b↓6⊗b↓5⊗b↓4⊗b↓3⊗b↓2⊗b↓1⊗b↓0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗\\⊗⊗\\\cr
B↓6⊗\\⊗0⊗b↓6⊗b↓5⊗b↓4⊗b↓3⊗b↓2⊗b↓1⊗b↓0⊗0⊗0⊗0⊗0⊗0⊗0⊗\\⊗⊗\\\cr
B↓5⊗\\⊗0⊗0⊗b↓6⊗b↓5⊗b↓4⊗b↓3⊗b↓2⊗b↓1⊗b↓0⊗0⊗0⊗0⊗0⊗0⊗\\⊗⊗\\\cr
B↓4⊗\\⊗0⊗0⊗0⊗b↓6⊗b↓5⊗b↓4⊗b↓3⊗b↓2⊗b↓1⊗b↓0⊗0⊗0⊗0⊗0⊗\\⊗⊗\\\cr
B↓3⊗\\⊗0⊗0⊗0⊗0⊗b↓6⊗b↓5⊗b↓4⊗b↓3⊗b↓2⊗b↓1⊗b↓0⊗0⊗0⊗0⊗\\⊗c↓4↑3/b↓6↑5⊗\\⊗D↓3\cr
B↓2⊗\\⊗0⊗0⊗0⊗0⊗0⊗b↓6⊗b↓5⊗b↓4⊗b↓3⊗b↓2⊗b↓1⊗b↓0⊗0⊗0⊗\\⊗c↓4↑3/b↓6↑5⊗\\⊗D↓2\cr
B↓1⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗b↓6⊗b↓5⊗b↓4⊗b↓3⊗b↓2⊗b↓1⊗b↓0⊗0⊗\\⊗c↓4↑3/b↓6↑5⊗\\⊗D↓1\cr
B↓0⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗b↓6⊗b↓5⊗b↓4⊗b↓3⊗b↓2⊗b↓1⊗b↓0⊗\\⊗c↓4↑3/b↓6↑5⊗\\⊗D↓0\cr
C↓5⊗\\⊗0⊗0⊗0⊗0⊗c↓4⊗c↓3⊗c↓2⊗c↓1⊗c↓0⊗0⊗0⊗0⊗0⊗0⊗\\⊗⊗\\\cr
C↓4⊗\\⊗0⊗0⊗0⊗0⊗0⊗c↓4⊗c↓3⊗c↓2⊗c↓1⊗c↓0⊗0⊗0⊗0⊗0⊗\\⊗⊗\\\cr
C↓3⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗c↓4⊗c↓3⊗c↓2⊗c↓1⊗c↓0⊗0⊗0⊗0⊗\\⊗⊗\\\cr
C↓2⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗c↓4⊗c↓3⊗c↓2⊗c↓1⊗c↓0⊗0⊗0⊗\\⊗⊗\\\cr
C↓1⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗c↓4⊗c↓3⊗c↓2⊗c↓1⊗c↓0⊗0⊗\\⊗d↓2↑2b↓6↑4/c↓4↑5⊗\\⊗E↓1\cr
C↓0⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗c↓4⊗c↓3⊗c↓2⊗c↓1⊗c↓0⊗\\⊗d↓2↑2b↓6↑4/c↓4↑5⊗\\⊗E↓0\cr
D↓3⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗d↓2⊗d↓1⊗d↓0⊗0⊗0⊗0⊗\\⊗⊗\\\cr
D↓2⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗d↓2⊗d↓1⊗d↓0⊗0⊗0⊗\\⊗⊗\\\cr
D↓1⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗d↓2⊗d↓1⊗d↓0⊗0⊗\\⊗⊗\\\cr
D↓0⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗d↓2⊗d↓1⊗d↓0⊗\\⊗e↓2↑2c↓4↑2/d↓2↑3b↓6↑2⊗\\⊗F↓0\cr
E↓1⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗e↓1⊗e↓0⊗0⊗\\⊗⊗\\\cr
E↓0⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗e↓1⊗e↓0⊗\\⊗⊗\\\cr
F↓0⊗\\⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗0⊗f↓0⊗\\⊗⊗\\\cr}
\vfill
% Appendix A (Tables of numerical quantities)
\runningrighthead{TABLES OF NUMERICAL QUANTITIES}
\section{}
\eject % eject the previous page
\runninglefthead{APPENDIX A}
\acpmark{\chd}{\csec}
\setcount0 659
\titlepage\corners
\vskip 1cm plus 30pt minus 10pt
\hbox{\def\\{\hskip 1pt}\:=A\\P\\P\\E\\N\\D\\I\\X\hskip 10pt A}
\vfill
\rjustline{\:;TABLES OF}
\vskip 4pt
\rjustline{\:;NUMERICAL QUANTITIES}
\vfill
%folio 843 galley 10b (C) Addison-Wesley 1978	*
\tablehead{Table 1}
\vskip 6pt
\inxf{multiple-precision, table of constants}
\ctrline{\hbox expand 11pt{QUANTITIES THAT ARE FREQUENTLY
USED IN STANDARD SUBROUTINES}}
\ctrline{\hbox expand 12pt{AND IN ANALYSIS OF COMPUTER PROGRAMS
(40 DECIMAL PLACES)}}
\ninepoint
\vskip 6pt \hrule \vskip 5pt
\ctrline{$\vbox{\halign{$\hfill#=\null$⊗#⊗$#$\cr
\sqrt{2}		⊗1.41421 35623 73095 04880 16887 24209 69807 85697⊗-\cr
\sqrt{3}		⊗1.73205 08075 68877 29352 74463 41505 87236 69428⊗+\cr
\sqrt{5}		⊗2.23606 79774 99789 69640 91736 68731 17623 54406⊗+\cr
\sqrt{10}		⊗3.16227 76601 68379 33199 88935 44432 71853 37196⊗-\cr
\spose{\raise4.5pt\hbox{\hskip2.3pt$\scriptscriptstyle3$}}
\sqrt{2}		⊗1.25992 10498 94873 16476 72106 07278 22835 05703⊗-\cr
\spose{\raise4.5pt\hbox{\hskip2.3pt$\scriptscriptstyle3$}}
\sqrt{3}		⊗1.44224 95703 07408 38232 16383 10780 10958 83919⊗-\cr
\spose{\raise4.5pt\hbox{\hskip1.3pt$\scriptscriptstyle4$}}
\sqrt{2}		⊗1.18920 71150 02721 06671 74999 70560 47591 52930⊗-\cr
\ln 2			⊗0.69314 71805 59945 30941 72321 21458 17656 80755⊗+\cr
\ln 3			⊗1.09861 22886 68109 69139 52452 36922 52570 46475⊗-\cr
\ln 10			⊗2.30258 50929 94045 68401 79914 54684 36420 76011⊗+\cr
1/\!\ln 2		⊗1.44269 50408 88963 40735 99246 81001 89213 74266⊗+\cr
1/\!\ln 10		⊗0.43429 44819 03251 82765 11289 18916 60508 22944⊗-\cr
π			⊗3.14159 26535 89793 23846 26433 83279 50288 41972⊗-\cr
1\deg=π/180		⊗0.01745 32925 19943 29576 92369 07684 88612 71344⊗+\cr
1/π			⊗0.31830 98861 83790 67153 77675 26745 02872 40689⊗+\cr
π↑2			⊗9.86960 44010 89358 61883 44909 99876 15113 53137⊗-\cr
\sqrt{π} = \Gamma(1/2)	⊗1.77245 38509 05516 02729 81674 83341 14518 27975⊗+\cr
\Gamma (1/3)		⊗2.67893 85347 07747 63365 56929 40974 67764 41287⊗-\cr
\Gamma (2/3)		⊗1.35411 79394 26400 41694 52880 28154 51378 55193⊗+\cr
e			⊗2.71828 18284 59045 23536 02874 71352 66249 77572⊗+\cr
1/e			⊗0.36787 94411 71442 32159 55237 70161 46086 74458⊗+\cr
e↑2			⊗7.38905 60989 30650 22723 04274 60575 00781 31803⊗+\cr
\gamma			⊗0.57721 56649 01532 86060 65120 90082 40243 10422⊗-\cr
\ln π			⊗1.14472 98858 49400 17414 34273 51353 05871 16473⊗-\cr
\phi			⊗1.61803 39887 49894 84820 45868 34365 63811 77203⊗+\cr
e↑\gamma		⊗1.78107 24179 90197 98523 65041 03107 17954 91696⊗+\cr
e↑{π/4}			⊗2.19328 00507 38015 45655 97696 59278 73822 34616⊗+\cr
\sin 1			⊗0.84147 09848 07896 50665 25023 21630 29899 96226⊗-\cr
\cos 1			⊗0.54030 23058 68139 71740 09366 07442 97660 37323⊗+\cr
-\zeta↑\prime(2)	⊗0.93754 82543 15843 75370 25740 94567 86497 78979⊗-\cr
\zeta (3)		⊗1.20205 69031 59594 28539 97381 61511 44999 07650⊗-\cr
\ln \phi		⊗0.48121 18250 59603 44749 77589 13424 36842 31352⊗-\cr
1/\!\ln \phi		⊗2.07808 69212 35027 53760 13226 06117 79576 77422⊗-\cr
-\ln\ln 2		⊗0.36651 29205 81664 32701 24391 58232 66946 94543⊗-\cr
}}$}
\vskip3pt
\hrule
%folio 843 galley 10b (C) Addison-Wesley 1978	*
\eject
\tablehead{Table 2}
\vskip 6pt
\ctrline{\hbox expand 11pt{QUANTITIES 
THAT ARE FREQUENTLY USED IN STANDARD SUBROUTINES}}
\ctrline{\hbox expand 12pt{AND IN ANALYSIS OF COMPUTER PROGRAMS (44 \α{OCTAL} PLACES)}}
\ninepoint
\vfill
\ctrline{The names at the left of the equal signs are given in decimal notation.}
\vfill \hrule \vskip 5pt
\ctrline{\baselineskip10.93pt$\vbox{\halign{$\hfill#=\null$⊗\hfill\it#\cr
0.1			⊗0.06314 63146 31463 14631 46314 63146 31463 14631 4632\cr
0.01			⊗0.00507 53412 17270 24365 60507 53412 17270 24365 6051\cr
0.001			⊗0.00040 61115 64570 65176 76355 44264 16254 02030 4467\cr
0.0001			⊗0.00003 21556 13530 70414 54512 75170 33021 15002 3522\cr
0.00001			⊗0.00000 24761 32610 70664 36041 06077 17401 56063 3442\cr
0.000001		⊗0.00000 02061 57364 05536 66151 55323 07746 44470 2603\cr
0.0000001		⊗0.00000 00153 27745 15274 53644 12741 72312 20354 0215\cr
0.00000001		⊗0.00000 00012 57143 56106 04303 47374 77341 01512 6333\cr
0.000000001		⊗0.00000 00001 04560 27640 46655 12262 71426 40124 2174\cr
0.0000000001		⊗0.00000 00000 06676 33766 35367 55653 37265 34642 0163\cr
\sqrt{2}		⊗1.32404 74631 77167 46220 42627 66115 46725 12575 1744\cr
\sqrt{3}		⊗1.56663 65641 30231 25163 54453 50265 60361 34073 4222\cr
\sqrt{5}		⊗2.17067 36334 57722 47602 57471 63003 00563 55620 3202\cr
\sqrt{10}		⊗3.12305 40726 64555 22444 02242 57101 41466 33775 2253\cr
\spose{\raise4.5pt\hbox{\hskip2.3pt$\scriptscriptstyle3$}}
\sqrt{2}		⊗1.20505 05746 15345 05342 10756 65334 25574 22415 0303\cr
\spose{\raise4.5pt\hbox{\hskip2.3pt$\scriptscriptstyle3$}}
\sqrt{3}		⊗1.34233 50444 22175 73134 67363 76133 05334 31147 6012\cr
\spose{\raise4.5pt\hbox{\hskip1.3pt$\scriptscriptstyle4$}}
\sqrt{2}		⊗1.14067 74050 61556 12455 72152 64430 60271 02755 7314\cr
\ln 2			⊗0.54271 02775 75071 73632 57117 07316 30007 71366 5364\cr
\ln 3			⊗1.06237 24752 55006 05227 32440 63065 25012 35574 5534\cr
\ln 10			⊗2.23273 06735 52524 25405 56512 66542 56026 46050 5071\cr
1/\!\ln 2		⊗1.34252 16624 53405 77027 35750 37766 40644 35175 0435\cr
1/\!\ln 10		⊗0.33626 75425 11562 41614 52325 33525 27655 14756 0622\cr
π			⊗3.11037 55242 10264 30215 14230 63050 56006 70163 2112\cr
1\deg=π/180		⊗0.01073 72152 11224 72344 25603 54276 63351 22056 1155\cr
1/π			⊗0.24276 30155 62344 20251 23760 47257 50765 15156 7007\cr
π↑2			⊗11.67517 14467 62135 71322 25561 15466 30021 40654 3410\cr
\sqrt{π} = \Gamma(1/2)	⊗1.61337 61106 64736 65247 47035 40510 15273 34470 1776\cr
\Gamma (1/3)		⊗2.53347 35234 51013 61316 73106 47644 54653 00106 6605\cr
\Gamma (2/3)		⊗1.26523 57112 14154 74312 54572 37655 60126 23231 0245\cr
e			⊗2.55760 52130 50535 51246 52773 42542 00471 72363 6166\cr
1/e			⊗0.27426 53066 13167 46761 52726 75436 02440 52371 0336\cr
e↑2			⊗7.30714 45615 23355 33460 63507 35040 32664 25356 5022\cr
\gamma			⊗0.44742 14770 67666 06172 23215 74376 01002 51313 2552\cr
\ln π			⊗1.11206 40443 47503 36413 65374 52661 52410 37511 4606\cr
\phi			⊗1.47433 57156 27751 23701 27634 71401 40271 66710 1501\cr
e↑\gamma		⊗1.61772 13452 61152 65761 22477 36553 53327 17554 2126\cr
e↑{π/4}			⊗2.14275 31512 16162 52370 35530 11342 53525 44307 0217\cr
\sin 1			⊗0.65665 24436 04414 73402 03067 23644 11612 07474 1451\cr
\cos 1			⊗0.42450 50037 32406 42711 07022 14666 27320 70675 1232\cr
-\zeta↑\prime(2)	⊗0.74001 45144 53253 42362 42107 23350 50074 46100 2771\cr
\zeta (3)		⊗1.14735 00023 60014 20470 15613 42561 31715 10177 0662\cr
\ln \phi		⊗0.36630 26256 61213 01145 13700 41004 52264 30700 4065\cr
1/\!\ln \phi		⊗2.04776 60111 17144 41512 11436 16575 00355 43630 4065\cr
-\ln\ln 2		⊗0.27351 71233 67265 63650 17401 56637 26334 31455 5701\cr
}}$}
\vskip3pt
\hrule
\eject
%folio 845 galley 12 (C) Addison-Wesley 1978	*
\tenpoint
Several of the values in Tables 1 and 2 were computed by John W. \α{Wrench},↔Jr.
For high-precision values of constants not
found in this list, see J. \α{Peters}, {\sl Ten Place Logarithms
of the Numbers from 1 to 100000\/}, Appendix to Volume 1
(New York: F. Ungar Publ.\ Co., 1957); and {\sl Handbook of Mathematical
Functions},
ed.\ by M. \α{Abramowitz} and I. A. \α{Stegun} (Washington, D.C.: U.S.
Gov't Printing Office, 1964), Chapter 1.

A table of Bernoulli numbers through $B↓{250}$ appears in a paper
by D. E. \α{Knuth} and T. J. \α{Buckholtz}, {\sl Math.\ Comp.\ \bf 21} (1967),
663--688.

\vfill
\tablehead{Table 3}
\vskip 6pt
\ctrline{\hbox expand 8pt{VALUES OF HARMONIC NUMBERS\null, BERNOULLI NUMBERS\null,}}
\ctrline{\hbox expand 11pt{AND FIBONACCI NUMBERS\null, FOR SMALL VALUES OF $n$}}
\inxf{harmonic numbers}
\inx{Bernoulli numbers}
\inx{Fibonacci numbers}
\ninepoint
\vskip 6pt \hrule \vskip 7pt
\hbox{$\vbox{\halign to size{\quad\hfill#\tabskip 0pt plus 100pt
⊗\hfill#\tabskip 0pt⊗#\hfill\tabskip 0pt plus 100pt
⊗$\hfill#$\tabskip 0pt⊗#\hfill\tabskip 0pt plus 100pt
⊗\hfill#⊗\hfill#\quad\tabskip0pt\cr
$n$\hskip 2pt⊗$H↓n\hskip-3pt$⊗⊗B↓n\hskip-3pt⊗⊗$F↓n$\9⊗$n$\hskip 2pt\cr
\noalign{\vskip 3pt}
0⊗0⊗⊗1⊗⊗0⊗0\cr
1⊗1⊗⊗-1/⊗2⊗1⊗1\cr
2⊗3/⊗2⊗1/⊗6⊗1⊗2\cr
3⊗11/⊗6⊗0⊗⊗2⊗3\cr
4⊗25/⊗12⊗-1/⊗30⊗3⊗4\cr
5⊗137/⊗60⊗0⊗⊗5⊗5\cr
6⊗49/⊗20⊗1/⊗42⊗8⊗6\cr
7⊗363/⊗140⊗0⊗⊗13⊗7\cr
8⊗761/⊗280⊗-1/⊗30⊗21⊗8\cr
9⊗7129/⊗2520⊗0⊗⊗34⊗9\cr
10⊗7381/⊗2520⊗5/⊗66⊗55⊗10\cr
11⊗83711/⊗27720⊗0⊗⊗89⊗11\cr
12⊗86021/⊗27720⊗-691/⊗2730⊗144⊗12\cr
13⊗1145993/⊗360360⊗0⊗⊗233⊗13\cr
14⊗1171733/⊗360360⊗7/⊗6⊗377⊗14\cr
15⊗1195757/⊗360360⊗0⊗⊗610⊗15\cr
16⊗2436559/⊗720720⊗-3617/⊗510⊗987⊗16\cr
17⊗42142223/⊗12252240⊗0⊗⊗1597⊗17\cr
18⊗14274301/⊗4084080⊗43867/⊗798⊗2584⊗18\cr
19⊗275295799/⊗77597520⊗0⊗⊗4181⊗19\cr
20⊗55835135/⊗15519504⊗-174611/⊗330⊗6765⊗20\cr
21⊗18858053/⊗5173168⊗0⊗⊗10946⊗21\cr
22⊗19093197/⊗5173168⊗854513/⊗138⊗17711⊗22\cr
23⊗444316699/⊗118982864⊗0⊗⊗28657⊗23\cr
24⊗1347822955/⊗356948592⊗-236364091/⊗2730⊗46368⊗24\cr
25⊗34052522467/⊗8923714800⊗0⊗⊗75025⊗25\cr
}}$}
\vskip3pt
\hrule
\eject
\tenpoint
For any $x$, let $\dispstyle H↓x = \sum
↓{n≥1}\left({1\over n} - {1\over n + x}\right)$. Then
$$\baselineskip17pt\lineskip4pt
\eqalign{H↓{1/2} ⊗= 2 - 2 \ln 2,\cr
H↓{1/3} ⊗\textstyle= 3 - {1\over 2}π/\sqrt{3} - {3\over
2}\ln 3,\cr
H↓{2/3} ⊗\textstyle= {3\over 2} + {1\over 2}π/\sqrt{3}
- {3\over 2} \ln 3,\cr
H↓{1/4} ⊗\textstyle= 4 - {1\over 2}π - 3 \ln 2,\cr
H↓{3/4} ⊗\textstyle= {4\over 3} + {1\over 2}π - 3 \ln 2,\cr
H↓{1/5} ⊗\textstyle= 5 - {1\over 2}π\phi \sqrt{(2 + \phi)/5}
- {1\over 2}(3 - \phi )\ln 5 - (\phi - {1\over 2})\ln(2
+ \phi ),\cr
H↓{2/5} ⊗\textstyle= {5\over 2} - {1\over 2}π/\phi \sqrt{2
+ \phi } - {1\over 2}(2 + \phi )\ln 5 + (\phi - {1\over 2})\ln(2
+ \phi ),\cr
H↓{3/5} ⊗\textstyle= {5\over 3} + {1\over 2}π/\phi \sqrt{2
+ \phi } - {1\over 2}(2 + \phi )\ln 5 + (\phi - {1\over 2})\ln(2
+ \phi ),\cr
H↓{4/5} ⊗\textstyle= {5\over 4} + {1\over 2}π\phi \sqrt{(2 + \phi
)/5} - {1\over 2}(3 - \phi )\ln 5 - (\phi - {1\over 2}\ln(2
+ \phi ),\cr
H↓{1/6} ⊗\textstyle= 6 - {1\over 2}π\sqrt{3} - 2\ln 2 -
{3\over 2}\ln 3,\cr
H↓{5/6} ⊗\textstyle= {6\over 5} + {1\over 2}π\sqrt{3} -
2 \ln 2 - {3\over 2} \ln 3,\cr}$$
and, in general, when $0 < p < q$ (cf.\ exercise 1.2.9--19),
$$H↓{p/q} = {q\over p} - {1\over 2}π \cot {p\over q} π -
\ln 2q + 2\sum ↓{1≤n<q/2}\cos {2πnp\over q} \ln\sin {n\over
q} π.$$

\vfill
% Appendix B (Index to notations)	*
\lineskip0pt
\def\∀{\hfil\linebreak}
\def\medlp{\mathopen{\vcenter{\hbox{\:@\char'20}}}}
\def\medrp{\mathclose{\vcenter{\hbox{\:@\char'21}}}}
\def\¬{\lower 2.5pt\vbox to 12pt{}}
\def\\{\noalign{\vfill}}
\def\+#1{\baselineskip12pt\lower 12pt\hbox par 192pt{\¬#1\¬}}
\def\2#1 #2{$\vtop{\baselineskip20pt\hbox{#1}
\hbox to 192pt{\ctr{$#2$}}}$}
\def\∨#1{\lower 12pt\hbox{#1}}
\def\∪#1{\lower 20pt\hbox{#1}}
\runningrighthead{INDEX TO NOTATIONS}
\section{}
\eject % eject the previous page
\runninglefthead{APPENDIX B}
\acpmark{\chd}{\csec}
\titlepage\corners
\vskip 1cm plus 30pt minus 10pt
\hbox{\def\\{\hskip 1pt}\:=A\\P\\P\\E\\N\\D\\I\\X\hskip 10pt B}
\vfill
\rjustline{\:;INDEX TO NOTATIONS}
\vfill
\tenpoint\noindent In the following formulas, letters that are not further
qualified have the following significance:
$$\vbox{\halign{$\hfill#$⊗\qquad#\hfill\cr
j,k⊗integer-valued arithmetic expression\cr
m,n⊗nonnegative integer-valued arithmetic expression\cr
x,y,z⊗real-valued arithmetic expression\cr
f⊗real-valued function\cr}}$$

\vskip 6pt
\hbox to size{\hskip 8pc\vrule height 300pt\hfill\vrule\hskip 4pc}
\vskip-300pt
\vbox to 300pt{
\hbox to size{\bf\hfill\hbox to 4pc{\9\ctr{Section}}}
\hbox to size{\bf\hbox to 8pc{\ctr{Formal symbolism}\9}\hfill
Meaning\hfill\hbox to 4pc{\9\ctr{reference}}}
\vskip3pt\hrule\vskip6pt
\baselineskip0pt
\halign{\hbox to 8pc{\hfill$\dispstyle #$\9}⊗\hskip 6pt\hbox to 192pt{\¬#\¬\hfill}
\hskip 6pt⊗\hbox to 4pc{\9#\hfill}\cr
\blackslug⊗end of algorithm, program, or proof⊗1.1\cr\\
A↓n⊗the $n$th element of linear array $A$\cr\\
A↓{mn}⊗\+{the element in row $m$ and column $n$ of\∀ rectangular array $A$}\cr\\
A[n]⊗equivalent to $A↓n$⊗1.1\cr\\
A[m,n]⊗equivalent to $A↓{mn}$⊗1.1\cr\\
V←E⊗give variable $V$ the value of expression $E$⊗1.1\cr\\
U\xch V⊗interchange the values of variables $U$ and $V$⊗1.1\cr\\
(B\→E;\;E↑\prime)⊗\+{conditional expression:\xskip denotes $E$ if $B$ is true,
$E↑\prime$ if $B$ is false}⊗\∨{8.1}\cr\\
\delta↓{kj}⊗\α{Kronecker} delta:\xskip$(j=k\→1;\;0)$⊗1.2.6\cr\\
\chop to 12pt{\sum↓{R(k)}f(k)}⊗\+{sum of all $f(k)$ such that the var\-iable 
$k$ is an integer and
relation $R(k)$ is true}⊗\∨{1.2.3}\cr\\
\chop to 12pt{\prod↓{R(k)}f(k)}⊗\+{product of all $f(k)$ such that the var\-iable 
$k$ is an integer
and relation $R(k)$ is true}⊗\∨{1.2.3}\cr\\
\chop to 12pt{\min↓{R(k)}f(k)}⊗\+{minimum value of all
$f(k)$ such that the var\-iable 
$k$ is an integer and relation $R(k)$ is true}⊗\∨{1.2.3}\cr\\
\chop to 12pt{\max↓{R(k)}f(k)}⊗\+{maximum value of all
$f(k)$ such that the var\-iable 
$k$ is an integer and relation $R(k)$ is true}⊗\∨{1.2.3}\cr\\
}}  % finish the \halign and the \vbox
\eject % eject the previous page
\hbox to size{\hskip 8pc\vrule height 45pc\hfill\vrule\hskip 4pc}
\penalty1000\vskip-45pc
\vbox to 45pc{
\hbox to size{\bf\hfill\hbox to 4pc{\9\ctr{Section}}}
\hbox to size{\bf\hbox to 8pc{\ctr{Formal symbolism}\9}\hfill
Meaning\hfill\hbox to 4pc{\9\ctr{reference}}}
\vskip3pt\hrule\vskip6pt
\baselineskip0pt
\halign{\hbox to 8pc{\hfill$\dispstyle #$\9}⊗\hskip 6pt\hbox to 192pt{\¬#\¬\hfill}
\hskip 6pt⊗\hbox to 4pc{\9#\hfill}\cr
A↑T⊗\2{transpose of rectangular array $A$:}
{A↑T[j,k]=A[k,j]}⊗\∪{1.2.3}\cr\\
x↑y⊗$x$ to the $y$ power (when $x$ is positive)⊗1.2.2\cr\\
x↑k⊗\2{$x$ to the $k$th power:}
{\chop to 9pt{\medlp k≥0\→\prod↓{0≤j<k}x;\quad 1/x↑{-k}\medrp}}⊗\∪{1.2.2}\cr\\
x↑{\overline k}⊗\2{$x$ upper $k$:}
{\chop to 9pt{\medlp k≥0\→\prod↓{0≤j<k}(x+j);\quad 1/(x+k)↑{\overline{-k}}\,
\medrp}}⊗\∪{1.2.6}\cr\\
x↑{\underline k}⊗\2{$x$ lower $k$:}
{\chop to 9pt{\medlp k≥0\→\prod↓{0≤j<k}(x-j);\quad 1/(x-k)↑{\underline{-k}}
\medrp}}⊗\∪{1.2.6}\cr\\
n!⊗$n$ factorial:\xskip$n↑{\underline n}$⊗1.2.5\cr\\
f↑\prime(x)⊗derivative of $f$ at $x$⊗1.2.9\cr\\
f↑{\prime\prime}(x)⊗second derivative of $f$ at $x$⊗1.2.10\cr\\
f↑{(n)}(x)⊗$\vtop{\baselineskip20pt
\hbox{$n$th derivative:\xskip$\biglp n=0\→f(x);\;g↑\prime(x)\bigrp$,}
\hbox to 192pt{\hfill where $g(x)=f↑{(n-1)}(x)$}}
$⊗\∪{1.2.11.2}\cr\\
H↓n↑{(x)}⊗harmonic number of order $x$:\xskip\chop to 9pt{\sum↓{1≤k≤n}1/k↑x}⊗\!
1.2.7\cr\\
H↓n⊗harmonic number:\xskip $H↓n↑{(1)}$⊗1.2.7\cr\\
F↓n⊗\2{Fibonacci number:}
{(n≤1\→n;\;F↓{n-1}+F↓{n-2})}⊗\∪{1.2.8}\cr\\
B↓n⊗Bernoulli number⊗1.2.11.2\cr\\
X\cdot Y⊗\+{dot product of vectors $X=(x↓1,\ldotss,x↓n)$\∀
and $Y=(y↓1,\ldotss,y↓n)$:\xskip$x↓1y↓1+\cdotss+x↓ny↓n$}\cr\\
(\ldotsm a↓1a↓0.a↓{-1}\ldotsm)↓b⊗radix-$b$ positional notation:\xskip
$\sum↓k a↓kb↑k$⊗4.1\cr\\
\vtop{\baselineskip12pt
\hbox{(min $x↓1$, ave $x↓2$,}
\hbox to 8pc{\hfill max $x↓3$, dev $x↓4$)\9}}\hskip-5pt
⊗\baselineskip12pt\lower24pt\hbox par 192pt{a random variable having minimum\∀
value $x↓1$, average (``expected'') value $x↓2$,\∀
maximum value $x↓3$, standard deviation $x↓4$\¬}⊗\lower24pt\hbox{1.2.10}\cr\\
}}  % finish the \halign and the \vbox
\eject % eject the previous page
\hbox to size{\hskip 8pc\vrule height 45pc\hfill\vrule\hskip 4pc}
\penalty1000\vskip-45pc
\vbox to 45pc{
\hbox to size{\bf\hfill\hbox to 4pc{\9\ctr{Section}}}
\hbox to size{\bf\hbox to 8pc{\ctr{Formal symbolism}\9}\hfill
Meaning\hfill\hbox to 4pc{\9\ctr{reference}}}
\vskip3pt\hrule\vskip6pt
\baselineskip0pt
\halign{\hbox to 8pc{\hfill$\dispstyle #$\9}⊗\hskip 6pt\hbox to 192pt{\¬#\¬\hfill}
\hskip 6pt⊗\hbox to 4pc{\9#\hfill}\cr
\dispstyle{x\choose k}⊗binomial coefficient: $(k<0\→0;\;x↑{\underline k}/k!)$⊗\!
1.2.6\cr\\
\dispstyle{n\choose n↓1,n↓2,\ldotss,n↓m}⊗\+{multinomial coefficient (defined only
when $n=n↓1+n↓2+\cdots+n↓m$)}⊗\∨{1.2.6}\cr\\
\dispstyle{n\comb[] m}⊗\2{Stirling number of the first kind:}
{\chop to 9pt{\sum↓{0<k↓1<k↓2<\cdots<k↓{n-m}<n}k↓1k↓2\ldotsm k↓{n-m}}}⊗\∪
{1.2.6}\cr\\
\dispstyle{n\comb\{\} m}⊗\2{Stirling number of the second kind:}
{\chop to 15pt{\sum↓{1≤k↓1≤k↓2≤\cdots≤k↓{n-m}≤m}k↓1k↓2\ldotsm k↓{n-m}}}⊗\∪
{1.2.6}\cr\\
\def\bslash{\char'477 } % boldface slash (vol. 2 only)
\bslash x↓1,x↓2,\ldotss,x↓n\bslash⊗\2{continued fraction:}
{1\vcenter{\hbox{\:@\char'16}}\biglp x↓1+1/(x↓2+1/(\,\cdots+1/(x↓n)\ldotsm))\bigrp
}⊗\∪{4.5.3}\cr\\
((x))⊗sawtooth function⊗3.3.3\cr\\
|x|⊗absolute value of $x$:\xskip$(x≥0\→x;\;-x)$\cr\\
\|S\|⊗cardinal:\xskip the number of elements in set $S$\cr\\
\lfloor x\rfloor⊗floor of $x$, greatest integer function:\xskip
\chop to 6pt{\max↓{k≤x}k}⊗1.2.4\cr\\
\lceil x\rceil⊗ceiling of $x$, least integer function:\xskip
\chop to 6pt{\min↓{k≥x}k}⊗1.2.4\cr\\
\leftset a\relv R(a)\rightset⊗set of all $a$ such that the relation $R(a)$ is
true\cr\\
\{a↓1,\ldotss,a↓n\}⊗the set $\leftset a↓k\relv 1≤k≤n\rightset$\cr\\
\{x\}⊗\+{fractional part (used in contexts where a\∀ real value, not a set, is
implied):\xskip$x-\lfloor x\rfloor$}⊗\∨{3.3.3}\cr\\
[y,z)⊗half-open interval:\xskip$\leftset x\relv y≤x<z\rightset$\cr\\
\langle X↓n\rangle⊗\+{the infinite sequence $X↓0$, $X↓1$, $X↓2$, $\ldots$\∀(here the
letter $n$ is part of the symbolism)}⊗\∨{1.2.9}\cr\\
\log↓b x⊗\+{logarithm, base $b$, of $x$ (real positive $x$\∀and $b$, where
$b≠1$):\xskip the $y$ such that $x=b↑y$}⊗\∨{1.2.2}\cr\\
\lg x⊗binary logarithm:\xskip$\log↓2 x$⊗1.2.2\cr\\
\ln x⊗natural logarithm:\xskip$\log↓e x$⊗1.2.2\cr\\
}}  % finish the \halign and the \vbox
\eject % eject the previous page
\hbox to size{\hskip 8pc\vrule height 45pc\hfill\vrule\hskip 4pc}
\penalty1000\vskip-45pc
\vbox to 45pc{
\hbox to size{\bf\hfill\hbox to 4pc{\9\ctr{Section}}}
\hbox to size{\bf\hbox to 8pc{\ctr{Formal symbolism}\9}\hfill
Meaning\hfill\hbox to 4pc{\9\ctr{reference}}}
\vskip3pt\hrule\vskip6pt
\baselineskip0pt
\halign{\hbox to 8pc{\hfill$\dispstyle #$\9}⊗\hskip 6pt\hbox to 192pt{\¬#\¬\hfill}
\hskip 6pt⊗\hbox to 4pc{\9#\hfill}\cr
\exp x⊗exponential of $x$:\xskip$e↑x$⊗1.2.2\cr\\
x\mod y⊗mod function:\xskip$\biglp y=0\→x;\;x-y\lfloor x/y\rfloor\bigrp$⊗1.2.4\cr\\
u(x)\mod v(x)⊗\+{remainder of polynomial $u$ after division by polynomial $v$}⊗\∨
{4.6.1}\cr\\
x≡y\modulo z⊗relation of congruence:\xskip$x\mod z=y\mod z$⊗1.2.4\cr\\
j\rslash k⊗$j$ divides $k$:\xskip$k\mod j=0$⊗1.2.4\cr\\
\hbox{B}(x,y)⊗beta function⊗1.2.6\cr\\
\gamma⊗Euler's constant:\xskip$\lim↓{n→∞}(H↓n-\ln n)$⊗1.2.7\cr\\
\gamma(x,y)⊗incomplete gamma function⊗1.2.11.3\cr\\
\Gamma(x)⊗gamma function:\xskip$\lim↓{y→∞}\gamma(x,y)$⊗1.2.5\cr\\
\delta(x)⊗characteristic function of the integers⊗3.3.3\cr\\
e⊗base of natural logarithms:\xskip$\sum↓{n≥0}1/n!$⊗1.2.2\cr\\
\zeta(x)⊗zeta function:\xskip$\lim↓{n→∞}H↓n↑{(x)}$ (when $x>1$)⊗1.2.7\cr\\
\lscr(u)⊗leading coefficient of polynomial $u$⊗4.6\cr\\
l(n)⊗length of shortest addition chain for $n$⊗4.6.3\cr\\
\Lambda(n)⊗von Mangoldt's function⊗4.5.3\cr\\
\mu(n)⊗M\"obius function⊗4.5.2\cr\\
O\biglp f(n)\bigrp⊗big-oh of $f(n)$, as the variable $n→∞$⊗1.2.11.1\cr\\
O\biglp f(x)\bigrp⊗\+{big-oh of $f(x)$, for small values of the\∀ variable $x$ (or 
for $x$ in some specified range)}⊗\∨{1.2.11.1}\cr\\
\varphi(n)⊗Euler's totient function:\xskip\chop to 18pt{\sum↓{\scriptstyle0≤k<n
\atop\scriptstyle\gcd(k,n)=1}1}⊗1.2.4\cr\\
π⊗circle ratio:\xskip$\sum↓{n≥0}\,(-1)↑n/(2n+1)$\cr\\
\phi⊗golden ratio:\xskip${1\over2}\biglp 1+\sqrt 5\,\bigrp$⊗1.2.8\cr\\
\emptyset⊗empty set:\xskip$\leftset x\relv 0=1\rightset$\cr\\
∞⊗infinity:\xskip larger than any number\cr\\
}}  % finish the \halign and the \vbox
\eject % eject the previous page
\hbox to size{\hskip 8pc\vrule height 45pc\hfill\vrule\hskip 4pc}
\penalty1000\vskip-45pc
\vbox to 45pc{
\hbox to size{\bf\hfill\hbox to 4pc{\9\ctr{Section}}}
\hbox to size{\bf\hbox to 8pc{\ctr{Formal symbolism}\9}\hfill
Meaning\hfill\hbox to 4pc{\9\ctr{reference}}}
\vskip3pt\hrule\vskip6pt
\baselineskip0pt
\halign{\hbox to 8pc{\hfill$\dispstyle #$\9}⊗\hskip 6pt\hbox to 192pt{\¬#\¬\hfill}
\hskip 6pt⊗\hbox to 4pc{\9#\hfill}\cr
\det(A)⊗determinant of square matrix $A$⊗1.2.3\cr\\
\gcd(j,k)⊗\2{greatest common divisor of $j$ and $k$:}
{\chop to 9pt{\medlp j=k=0\→0;\;\max↓{d\rslash j,\,d\rslash k}d\medrp
}}⊗\∪{4.5.2}\cr\\
\lcm(j,k)⊗\2{least common multiple of $j$ and $k$:}
{\chop to 9pt{\medlp j=k=0\→0;\;\min↓{d>0,\,
j\rslash d,\,k\rslash d}d\medrp}}⊗\∪{4.5.2}\cr\\
\hbox{sign}(x)⊗sign of $x$:\xskip$\biglp x=0\→0;\;x/|x|\bigrp$\cr\\
\Pr\biglp S(n)\bigrp⊗\baselineskip12pt
\lower24pt\hbox par 192pt{probability that statement $S(n)$
is true, for ``random'' integers $n$ (here the letter $n$ is part of the
symbolism)\¬}⊗\lower24pt\hbox to 4pc{3.5, 4.2.4\hskip0pt plus10000pt minus10000pt
}\hskip-5pt\cr\\
\hbox{mean}(g)⊗\+{mean value of the probability distribution\∀represented by
generating function $g$:\xskip $g↑\prime(1)$}⊗\∨{1.2.10}\cr\\
\hbox{var}(g)⊗$\vtop{\baselineskip26pt
\hbox{\+{variance of the probability distribution\∀ represented by generating
function $g$:}}
\hbox to 192pt{\ctr{$\dispstyle
g↑{\prime\prime}(1)+g↑\prime(1)-g↑\prime(1)↑2$}}}$⊗\lower26pt\hbox{1.2.10}\cr\\
\hbox{deg}(u)⊗degree of polynomial $u$⊗4.6\cr\\
\hbox{cont}(u)⊗content of polynomial $u$⊗4.6.1\cr\\
\hbox{pp}\biglp u(x)\bigrp⊗primitive part of polynomial $u$⊗4.6.1\cr\\
\real(w)⊗real part of complex number $w$\cr\\
\imag(w)⊗imaginary part of complex number $w$\cr\\
\overline w⊗complex conjugate:\xskip$\real(w)-\imag(w)\,i$\cr\\
\.{\char'40}⊗one blank space⊗1.3.1\cr\\
\rA⊗register A (accumulator) of \MIX⊗1.3.1\cr\\
\rX⊗register X (extension) of \MIX⊗1.3.1\cr\\
\rI1,\ldotss,\rI6⊗(index) registers I1, $\ldotss$, I6 of \MIX⊗1.3.1\cr\\
\hbox{rJ}⊗(jump) register J of \MIX⊗1.3.1\cr\\
\.{(L:R)}⊗partial field of \MIX\ word, $0≤\.L≤\.R≤5$⊗1.3.1\cr\\
\.{OP ADDRESS,I(F)}⊗notation for \MIX\ instruction⊗1.3.1, 1.3.2\hskip-3.5pt\cr\\
u⊗unit of time in \MIX⊗1.3.1\cr\\
\.*⊗``self'' in \.{MIXAL}⊗1.3.2\cr\\
\hbox to 8pc{\hfill\.{0F}, \.{1F}, \.{2F}, $\ldotss$, \.{9F}\9}\hskip-5pt
⊗``forward'' local symbol in \.{MIXAL}⊗1.3.2\cr\\
\hbox to 8pc{\hfill\.{0B}, \.{1B}, \.{2B}, $\ldotss$, \.{9B}\9}\hskip-5pt
⊗``backward'' local symbol in \.{MIXAL}⊗1.3.2\cr\\
\hbox to 8pc{\hfill\.{0H}, \.{1H}, \.{2H}, $\ldotss$, \.{9H}\9}\hskip-5pt
⊗``here'' local symbol in \.{MIXAL}⊗1.3.2\cr\\
\oplus\,\ominus\,\otimes\,\odiv⊗rounded or special operations⊗4.2.1\cr\\
}}  % finish the \halign and the \vbox
\end % eject previous page and terminate