Concrete Mathematics, Second Edition

by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik (Reading, Massachusetts: Addison-Wesley, 1994), xiii+657pp.
ISBN 0-201-55802-5

Chinese translation by Lai FeiPei, Ju Ti Shu Xue (Taipei: Dong Hua Publishing Co., 1990), xv+731pp.
Chinese translation by Chen YanWen, Ju Ti Shu Xue (Taipei: Ru Lin Publishing Co., 1991), xii+695pp.
Chinese translation by Xingu Zhuang, Ju Ti Shu Xue (Xi'an: Xi An Dian Zi Ke Ji Da Xue Chu Ban She, 1992), xii+539pp.
Chinese translation of the second edition by Zhang Mingyao and Zhang Fan, Ju Ti Shu Xue -- Ji Suan Ji Ke Xue Ji Chu (Beijing: Turing Books, 2013), xii+564pp.
Italian translation edited by Giovanni Monegato, Matematica Discreta (Milan: Editore Ulrico Hoepli, 1992), xviii+607pp.
Japanese translation by Makoto Arisawa, Michiaki Yasumura, Tatsuya Hagino, and Kiyoshi Ishihata, Kompyuta no Suugaku (Tokyo: Kyoritsu-Shuppan, 1993), xvi+606pp.
Japanese translation of the second edition by Makoto Arisawa, Michiaki Yasumura, Tatsuya Hagino, and Kiyoshi Ishihata, Kompyuta no Suugaku (Tokyo: Kyoritsu-Shuppan, 2020), xvi+624pp.
Portuguese translation by Valéria de Magalhães Iorio, Matemática Concreta (Rio de Janeiro: Livros Técnicos e Cientíicos Editora, 1995), xii+477pp.
Russian translation by A. B. Khodulev and B. B. Pokhodzei, with foreword by V. Arnol'd, Конкретная математика (Moscow: Mir, 1999), 703pp.; see also (Moscow: Binom, 2009).
Polish translation by P. Chrzastowski, A. Czumaj, L. Gasieniec, and M. Raczunac, Matematyka Konkretna (Warszawa: Polskie Wydawnictwa Naukowe, 1996), 718pp.
Hungarian translation by S. Fridli, J. Gonda, A. Kovács, L. Lakatos, and Cs. Láng, Konkrét matematika (Budapest: Műszaki Könyvkiadó, 1998), xvi+647pp.
Spanish translation (Addison-Wesley Spain and Universidad Autónoma de Madrid), in preparation.
French translation by Alain Denise, Mathématiques concrètes (Paris: International Thomson Publishing, 1998), xiv+688pp.; now distributed by Éditions Vuibert.
Greek translation by Christos A. Kapoutsis (Athens: Klidarithmos, 2011), 640pp.
Korean translation by Ryu Gwang, 구체 수학 (Seoul: Insight Press, 2018), 816pp.
Croatian translation (Zagreb: Golden Marketing), in preparation.
Macedonian translation (Skopje: Ars Lamina), in preparation.

Introduction to the mathematics that supports advanced computer programming and the analysis of algorithms. An indispensable text and reference not only for computer scientists --- the authors themselves rely heavily upon it --- but for serious users of mathematics in virtually every discipline. Based on the course Concrete Mathematics taught by Knuth at Stanford University from 1970--1989. The second edition includes important new material about the revolutionary Gosper-Zeilberger algorithm for mechanical summation. Complete answers are provided for more than 500 exercises.

It is to be hoped that this book succeeds in convincing many educators, not only in computer science but also in mathematics, that courses like this pay off! -- J. H. Van Lint, International Reviews on Education (1990)
It is always a pleasure to look again into this book full of carefully explained and enthusiastically presented mathematics. -- Volker Strehl, Mathematical Reviews (1997)
Although designed as a textbook, the book is also a valuable reference, and is loaded with useful results, especially in the exercises (all of which are solved in the back of the book). This is my go-to book for any problem dealing with binomial coefficients, Fibonacci numbers, harmonic numbers, or evaluation of finite sums. -- Allen Stenger, MAA Reviews (2010)
This is one of those books you keep forever, purely for its utility. -- Naomi Novik, review on (2010)

Major topics include

Available from the publisher, Addison-Wesley Publishing Company.


For a list of corrections to known errors in the pre-1998 printings of the second edition, you may download the 1994--1997 errata file in compressed PostScript format (74K bytes). This file was generated by the TeX file errata97.tex in directory using special macros and fonts found in that directory. Errata lists for various printings of the first edition can also be found there. See also the list of errors corrected between January 1998 and May 2013, which is in HTML format.

And here is a list of all nits that have been picked so far since the 27th printing (May 2013), not counting the important replacement pages for the changes in the 34th printing (January 2022) because of a correction to the definition of Bernoulli numbers. An asterisk (*) marks significant technical errors that are not merely typographical:

page 1, line 2 before the illustration
change 'Edouard' to 'Édouard'
page 8, line 8
change 'only two regions per line' to 'only two regions per bent line'
page 28, line 4
change "quicksort" to "quicksort [209, Algorithm 5.2.2Q]"
*page 30, lines 8, 11, 17, and 25
change 'associative law' to 'pairing law'
*page 31, line 8 from the bottom
change 'associative' to 'pairing'
*page 33, line 11
change 'associative law' to 'pairing law'
*page 34, line 2 from the bottom
change 'associative law' to 'pairing law'
*page 41, line 3
change 'associative law' to 'pairing law'
*page 60, line 8 from the bottom
change 'associative' to 'pairing'
*page 61, line 3
change 'associative law' to 'pairing law'
*page 63, line 4 of exercise 11
change 'associative' to 'pairing'
*page 64, line 4 of exercise 25
change 'associative' to 'pairing'
page 75, line 8
use a bigger [ before $m\in$ and a bigger ] after $/k)$
page 77, line 2
change 'a real number' to 'a positive real number'
page 87, display on lines 8--10
change '$k<n$' to '$k<a^2$' under the first $\sum$; and append ', integer $a$' at the end of the formula.
page 101, line 1 of exercise 52
*change 'nonnegative real numbers $\alpha$ and $\beta$' to 'real numbers $\alpha$ and $\beta$ with $\alpha>0$ and $\alpha+\beta>0$'
*page 101, line 6 of exercise 52
change 'rational' to 'distinct'
*page 101, last line of exercise 52
change '$2^{k-1}$' to '$2^{m-k}$'
pages 107 (bottom) and 108 (top)
change 'Suppose there were only finitely many ... None of the' to: 'Consider any finite set of primes, say $\{P_1,P_2,\ldots,P_k\}$. Then, said Euclid, we should think about the number
$$M=P_1\cdot P_2\cdot\ldots\cdot P_k+1.$$
None of the'
page 108, line 7 (in the 27th printing only)
change 'primes numbers' to 'prime numbers'
page 108, a new graffito
"Oh, I see: The formula $e_1e_2\ldots e_{n-1}$ means $\prod_{k=1}^{n-1}e_k$, so it is equal to $1$ (an empty product) when $n=1$."
page 109, line 13
change '1984' to '1985'
page 114, line 11
change "progression:" to "progressions: When $n>0$ we have"
page 128, line 12 from the bottom
change "page 274" to "page 290"
page 151, line 15 (the sequence for script-P sub 5)
change the second occurrence of 2/5 to 3/5
page 167, line 10 (the displayed formula)
change "integer $m\ge0$" to "integer $m$"
page 178, line 10 from the bottom (the displayed formula)
change $m\ge0$ to $m\ge1$
page 178, line 4 from the bottom (the displayed table)
delete the entry for $m=0$
page 179, line 9 from the bottom
change $m\ge6$ to $m\ge7$
page 192, lines 2 and 3 from the bottom
change 'rule (5.45)' to 'rule (5.44)', and change 'way:' to 'way using (5.40):'
page 201, add new graffito in bottom half of page:
News flash: $\ln\Bscr_t(z)=\sum_{k\ge1}{tk\choose k}z^k/(tk)$; $\ln\Escr_t(z)=\sum_{k\ge1}(tk)^{k-1}z^k/k!$.
page 213, lines 6, 7, and 8
replace these lines by: general sum $\sum_{k\le n}{n-k\choose k}z^k$ considered in (5.74); and that sum leads, for example, to a closed-form expression for $$F\Biggl({n+1,-n\over 1/2}\Biggm|{-z\over4}\Biggr) = \sum_{k\ge0}{n+k\choose n-k}z^k = \sum_{k\le n}{2n-k\choose k}z^{n-k}.$$
page 214, line 10
change "integer n\ge0.$$ When" to "integer n\ge0,$$ if we replace $(a,b,c,k)$, respectively by $(a+n-1,b+n-1,n,k-n)$. When"
page 283, and 43 other pages!
see the replacement pages mentioned above
page 287, line 3
change $\sum_{n\ge0}$ to $\sum_{n\ge1}$
page 288, line 4
change 'sum of $n$th powers' to 'sum of $m$th powers'
page 305, line 14
omit '(This is "Halphen's theorem" [174].)'
page 306, line 1
change 'Thus' to 'Thus—as implicitly observed by Halphen [174]—'
page 317, last line of exercise 76
change '$n$ is even' to '$n>0$ is even'
page 373, line 6
change $P(z)$ to $\widehat P(z)$
page 428, line 9
enclose $\alpha^{\alpha n}(1-\alpha)^{(1-\alpha)n}$ in parentheses
page 429, exercise 8.22
change "law of conditional expectations and variances" to "law of total variance"
page 450, equation (9.27)
add the qualification ", if $f(n)$ is never zero"
page 477, line 10
change $R_4(n)$ to $R_4(n)+O(n^{-5})$
page 481, lines 8, 9, 10 from the bottom
change "(The value ... constant exists.)" to: "(Stirling's original formula was actually a bit different; (9.91) is de Moivre's modification [76]. Stirling [343, p. 137] stated without proof that $e^\sigma=\sqrt{2\pi}$; we aren't quite ready to prove that fact.)"
page 481, bottom line
change 'approximated' to 'approximated via Euler's formula'
page 488, bottom line
change $k>n^{1/2+\epsilon$ to $\vert k\vert>n^{1/2+\epsilon$
page 490, lines 1 and 2 of exercise 21
change $O(\log n)^{-2}$ to $O(\bigl((\log n)^{-2}\bigr)$ and $O(\log n)^{-3}$ to $O(\bigl((\log n)^{-3}\bigr)$
page 504, last line of answer 2.25
change $\sum_{k\in K}1=\#K$ to $\sum_{k\in K}c=c\#K$
page 514, line 4 from the bottom
change ‘irrational’ to ‘a repeated
*page 514, bottom line and top of page 515
change 'rational examples' to 'examples with distinct $\alpha$'s'
page 517, line 1
change $\rho(2k+1)$ to $\rho(2k-1)$
page 525, new copy for answer 4.67
M. Szegedy proved this conjecture for all large n; then R. Balasubramanian and K. Soundararajan found a complete solution. See [95, pp. 78-79], [348], [55], and [19'].
page 525, new graffito for answer 4.69
Late-breaking news: The $10,000 prize has been won! See Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao, Journal of the American Mathematical Society, vol. 31 (2018), pages 65–105.
page 549, new graffito for answer 5.114
Late-breaking news: W. Zudilin has found an explicit expression for each $c_n^{(m)}$ as a summation of binomial coefficients; see Electronic Journal of Combinatorics, vol. 11 (2004), #R22, pages 1–8.
page 552, new answer 6.32
$\sum_{0\le k\le m}k{n+k\brace k}={m+n+1\brace k}-{n\brace-1}= {m+n+1\brace k}-[n=-1]$ generalizes (6.22) to negative $n$; and $\sum_{k\le n}{k\brace m}(m+1)^{n-k}={n+1\brace m+1}$ generalizes (6.20) to negative $m$, $n$.
*page 553, line 3 of answer 6.44
change $(m-k,m)$ to $(k,m-k)$
page 577, line 4 of answer 8.19
change "when $a,b\ge0$" to "when $X_1$ and $X_2$ are independent"
page 580, replacement for lines 2 and 3 of answer 8.31
$A=1$; $B=\half zA+\half zC$; $C=\half zB+\half zD$; $D=\half zC+\half zE$; $E=\half zD+\half zA$. (And the following lines are revised to fit this better approach.)
page 580, line 3 from the bottom
change $1/x^{98}(4-2x-x^2)$ to $1/(x^{98}(4-2x-x^2))$
page 593, in answer 9.19
change 2.928968256 to 2.928968258 and 3628712.4 to 3628800.05
page 604, marginal note for reference 2
change "42." to "42, 284."
page 605, new reference
19' R. Balasubramanian and K. Soundararajan, “On a conjecture of R. L. Graham,” Acta Arithmetica 75 (1996), 1--38.
page 607, in reference 44
change 'subpolynormal' to 'sub-polynormal'
page 610, in reference 83
change '147--223' to '145--223'
page 617, in reference 174
change '305.' to '306.' and '1876' to '1877'
page 621, in reference 235
change 'della' to 'delle'
page 628, in reference 330
change "" to "Since 2011,"
page 629, in reference 343
insert a crossreference to page 481 in the margin; and add this: Annotated translation by Ian Tweddle, Springer-Verlag, 2003.
page 638, replacement for associative law entry
associative law, 561
page 638, new entry
Balasubramanian, Ramachandran, 525, 605.
page 644, new entry
Ford, Kevin Barry, 525
page 644, Franel entry
Franel, Jérôme, 549, 614
page 645, Ron Graham entry
add page 605
page 646, new entry
Green, Ben Joseph, 525
page 647. new entry
Konyagin, Sergei Vladimirovich, 525
page 647, Lagrange entry
Lagrange, Joseph Louis (= De la Grange Tournier, Giuseppe Ludovico (= Luigi)), 470, 621, 635
page 647, Landau entry
change '622' to '621–622'
page 648, new entry
Maynard, James
page 650, Pacioli entry
Pacioli, Luca Bartolomeo de, 614
page 650, new entry
pairing law, 30, 33, 41, 61, 64
page 652, left column
regular expressions, 326
page 653, new entry
Soundararajan, Kannan, 525, 605.
page 655. new entry
Tao, Terence Chi-Shen, 525
page 655, new entry
Tweddle, Ian, 629
page 656, new entry
Zudilin, Wadim Valentinovich, 549

With these corrections, the authors hope that the book is now error-free. But (sigh) it probably isn't. Therefore I will gratefully deposit 0x$1.00 ($2.56) to the account of the first person who finds and reports anything that remains technically, historically, typographically, or politically incorrect. Please send suggested corrections to, or send snail mail to

Prof. D. Knuth
Computer Science Department
Gates Building 1B
Stanford University
Stanford, CA 94305-9015 USA.

In either case please include your postal address, so that I can mail an official certificate of deposit as a token of thanks for any improvements to which you have contributed.

I may not be able to read your message until many months have gone by, because I'm working intensively on The Art of Computer Programming. However, I promise to reply in due time.

DO NOT SEND EMAIL TO KNUTH-BUG EXCEPT TO REPORT ERRORS IN BOOKS! And if you do report an error via email, please do not include attachments of any kind; your message should be readable on brand-X operating systems for all values of X.

SPECIAL NOTE TO THE SPEAKERS OF FRENCH AND OTHER EXOTIC LANGUAGES: Numerous quotations and bibliographic citations found in this book have been copied verbatim from the original sources. If you believe you have found a typographic error, you must prove it by showing that the original was incorrectly transcribed; believe it or not, your language has changed over the years, just as English has.

Sample exams

Midterm and final exams from the last two times I taught this course at Stanford (1988 and 1989) are now available, together with the answers, as compressed PostScript files. The teaching assistants who helped to prepare these exams were Sheralyn Listgarten and Sridhar Raman (1988); Alan Hu and Robert Kennedy (1989).

Ron Graham's home page

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Oren Patashnik's home page

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