Submitting errata

If you notice any mistakes in the book, big or small, please send an email to the author at weissman AT ucsc DOT edu.

Please include the subject line ERRATUM (or ERRATA if plural, I guess!).

Before submitting your erratum, please look through the list below to see if it has already been caught.

When submitting errata, please include the following information:

All accepted errata will be acknowledged on this page, and also in future editions of the text. Thank you to all the colleagues and students who caught errata before the first edition was published! Thank you to Spencer Martin, at Cleveland State University, who found the first post-publication erratum. Thank you to Dr. Paul Stanford, at the University of Texas, Dallas, who found around 30 errata (!) and made excellent suggestions which I hope will be incorporated into the second edition someday.

Current errata (mathematical errors in red)


Chapter 0: Seeing arithmetic
Page 1 (darnit)
The word "myred" should be "red". Long story on this one: You might notice that some red text/dots in the text are more orange-ish than red. The reason is that screen-red (RGB) looks orange when printed. I thought I took care of this by creating a good print-red (in CMYK colorspace) which I called "myred". Indeed, I changed lots of red to "myred". But I missed some dots and text. And I accidentally changed the word "red" to "myred" on page 1. Whoops.

Page 3

The stacking-corners diagram is 13+1 dots by 13+1 dots, but the labels suggest that it is 14+1 dots by 14+1 dots. (Thanks to Spencer Martin, August 2017.)
Page 5
Marginnote, top-right: Zero is also typically considered a triangular number.
Page 7
Marginnote, first sentence: "triangles" should be "triangle". (Thanks to Chris Shelley, November 2017.)
Page 7
Paragraph below the big figure: Delete "on" in "Add only on..." (Thanks to Conner Jure, January 2018.)
Page 9
Just above Prop 0.13: "are form a" should be "are from a".
Page 14
Line 4: "they becomes second nature" should be "they become second nature". (Thanks to Erik Wallace, January 2018.)
Page 16
Line 2: "analagous" should be "analogous". (Thanks to Alberto Trombetta, January 2018.)
Page 16
The end-of-proof square comes a few lines too early in Proposition 0.27.
Page 16
Last line: "mutliple" should be "multiple". (Thanks to Alberto Trombetta, January 2018.)
Page 21
Exercise 24: By "When is T(N) even?" please understand "For which values of N is T(N) even?". (Thanks to Erik Wallace, January 2018.)

Page 21

Exercise 26: 6N should be 6 S(N). Also "perfect squares" should be called "squares". (Thanks to Patrick McDonald, January 2018.)
Page 21
Exercise 27: The parenthesis ) should be removed after "Notes". (Thanks to Chris Shelley, November 2017.)

Part I: Foundations


Chapter 1: The Euclidean algorithm
Page 34
Line 13: "equaions" should be "equation".

Page 36

Proposition 1.20: The integers a,b should both be nonzero.
Page 36
Proposition 1.20: The end-of-proof box is missing. At the end of the proof, the reader may verify that the given u,v are solutions of the equation au + bv = 0 for every n. (Thanks to Andrés Eduardo Caicedo, January 2017.)

Page 36

Theorem 1.21: The integers a,b should both be nonzero here too! The Theorem almost works when one (a or b) is zero, but not quite at the end. (Thanks to Andrés Eduardo Caicedo, December 2017.)
Page 39
The end-of-proof (QED) square is misplaced. It should be just above Problem 1.24.

Page 40

Corollary 1.25: The integers a,b should both be nonzero here too. (Thanks to Andrés Eduardo Caicedo, December 2017.)
Page 41
Proposition 1.27 and Corollary 1.29: The end-of-proof boxes are missing. (Thanks to Erik Wallace, January 2018.)

**Page 45**

Exercise 14: The last line should read "LCM(u^2, v^2) = LCM(u,v)^2". The final "squared" was forgotten.
Page 45
Exercise 19: The word "same" is missing, and the last line should read "look (geometrically) the same right-side-up". (Thanks to Jeffrey S. Haemer, January 2018.)

Page 45

Exercise 21: The question "How often...?" should be made more precise. If the tortoise begins jogging at time zero, describe all of the times at which the tortoise and hare cross paths (assuming they run forever at the same pace). (Thanks to Erik Wallace, January 2018.)

Chapter 2: Prime factorization
Page 49
Marginnote: A primality certificate typically refers to something more than we've described: a specific theorem and relevant parameters that rapidly guarantee primality.
Page 51
Figure 2.1, Marginnote: The record has been broken! News about the newest and largest Mersenne prime can be found at the homepage of the Great Internet Mersenne Prime Search. (Thanks to Andrés Eduardo Caicedo, January 2017.)
Page 53
The appearance of Li(x) and li(x) may be confusing. Li(x) is defined on the page, as the integral from 2 to x of (1 / log(t)) dt. The function li(x) is the integral from 0 to x of (1 / log(t)) dt. But this integral is a bit subtle due to the singularity of 1 / log(t) at t=1. The integral defining li(x) should be interpreted as the principal value. Or one may forget about this difficulty, and define li(x) to be Li(x) plus a constant, as in the footnote on this page. (Thanks to Andrés Eduardo Caicedo, December 2017.)
Page 54
Theorem 2.11 should really be attributed to Zhang-Maynard-PolyMath, as explained in the Historical Notes on p.71. (Thanks to Andrés Eduardo Caicedo, December 2017.)
Page 57
The exponent of 7 should be called f_7, and not f_5, in equations (2.1), (2.2), and (2.3).
Page 57
Last paragraph: For the proof of existence of prime decomposition, look back to page 48. (Thanks to Erik Wallace, February 2018.)
Page 62
Sidenote 19: Change "has" to "as" in "do not have 2 has a common factor". (Thanks to Jeffrey S. Haemer, January 2018.)
Page 63
Line 3: change "generators" to "generates". (Thanks to Andrés Eduardo Caicedo, December 2017.)
Page 63
Theorem 2.28: The proof is missing its end-of-proof box. (Thanks to Harrison Henningsen, February, 2018.)
Page 64
The reference for Jacobi's four-square and eight-square theorems is C.G.J. Jacobi, Fundamenta nove theoriae functionum ellipticarum, Königsberg (1829). (Thanks to Andrés Eduardo Caicedo, December 2017.)
Page 68
A period wandered away from 2N, at the bottom of the page.
Page 69
An update: by a 2015 paper of Pace P. Nielsen, it is now known that an odd perfect number must have at least 10 distinct prime factors, (Thanks to Andrés Eduardo Caicedo, December 2017.)

Page 71

In Footnote 40, the statement li(n) - Li(n) = log(2) is false. Instead, this should be li(n) - Li(n) = li(2), and li(2) is approximately 1.045, as stated in Footnote 8 on p.53. (Thanks to Andrés Eduardo Caicedo, December 2017.)
Page 72
Exercise 1: The first "and" should be deleted. (Thanks to Jeffrey S. Haemer, January 2018.)
Page 72
Exercise 3(b): Every element of T *greater than 1* can be factored as a product of irreducible elements. Or, you can read this as some mathematicians would, noting that 1 equals the "empty product". (Thanks to Jeffrey S. Haemer, January 2018.)

Page 73

Exercise 17: One must require n > 1 for this construction of amicable numbers to work! (Thanks to Andrés Eduardo Caicedo, January 2017.)
Page 73
Exercise 18(b): The period should go within the parenthesis at the end.

Chapter 3: Rational and constructible numbers
Page 76
Marginnote above figure: Change "one associate" to "one can associate". (Thanks to Andrés Eduardo Caicedo, January 2017.)
Page 76
Marginnote 3: Change "from" to "and" in "...both 1/0 from -1/0...". (Thanks to Harrison Henningsen, February, 2018.)

Page 80

Middle figure: the r^2/d should be r^2/d^2, inside both square roots. As a result, a 1/d could be factored out of the square roots to simplify the expression a bit.
Page 80
An end-of-proof box is missing at the bottom of the page. (Thanks to Andrés Eduardo Caicedo, January 2018.)

Page 81

Middle of page: "Expanding and multiplying through by cd" should be "expanding and multiplying through by c^2 d^2." (Thanks to Jeffrey S. Haemer, February 2018.)

Page 81

Ten lines from the bottom, the x^2 + y^2 should be a u^2 + v^2. (Thanks to Andrés Eduardo Caicedo, January 2018.)

Page 81

Sidenote 16: u should be a/c (not a/b) and v should be b/c. (Thanks to Jeffrey S. Haemer, February 2018.)
Page 82
Second paragraph, delete "must" in "must would". (Thanks to Andrés Eduardo Caicedo, January 2018.)

Page 83

Theorem 3.8 (Rational Root Theorem): The constants c0, ..., cd do not have to be positive, and in many important examples, they won't be! (Thanks to Junecue Suh, January 2018.)
Page 87
Figure 3.10, marginnote: The Ford circle at 7/3 is not depicted; instead the circle at the non-reduced fraction 4/2 is depicted, which sadly does not osculate any other circles. (Thanks to Andrés Eduardo Caicedo, January 2018.)

Page 88

Theorem 3.15: One must assume that the rational numbers a/b and c/d are distinct for the proof to go through as written. (Thanks to Andrés Eduardo Caicedo, January 2018.)

Page 90

Figure 3.15, marginnote: In fact, bx is not equal to 1. Place x within an absolute value: b |x| = 1. (Thanks to Andrés Eduardo Caicedo, January 2018.)
Page 90
Proposition 3.18: A remark... the proof only considers the Diophantine equation ay - bx = 1, whereas kissing fractions also arise from solutions to ay - bx = -1. But (x,y) is a solution to the first equation if and only if (-x, -y) is a solution to the second equation, and both (x,y) and (-x,-y) produce the same rational number y/x. (Thanks to Andrés Eduardo Caicedo, January 2018.)
Pages 90-95
I really should have mentioned Farey sequences by name, and in the historical notes.
Page 94
Last line: Oops.. it should read that the margin is too small to contain the proof, not that the proof is not too small to fit in the margin! Apologies to Fermat and Mr. Barnes, my high-school Latin teacher. (Thanks to Harrison Henningsen, February, 2018.)
Page 95
Sidenote 41: "...on can try..." should be "...one can try..."

Page 96

Exercise 2: The triples (x,y,z) should be required to be pairwise coprime, i.e. GCD(x,y) = GCD(y,z) = GCD(x,z) = 1. Otherwise a single triple easily yields infinitely many by scaling. (Thanks to Steven Gubkin, October 2017)
Page 96
Exercise 6(b): There's an extra period in "cosine.."

Chapter 4: Gaussian and Eisenstein integers
Page 98
The Gauss-inert and Eisenstein-inert primes are mean to be highlighted in blue. But perhaps they look black in print. This image needs some color-tuning.

Page 99

Sidenote: For the explanation to make sense, "integral domain" should have been defined. An "integral domain" is a ring in which xy = 0 implies x=0 or y=0.
Pages 101, 102
Problems 4.2 and 4.4: the word "Solution" and the end-of-solution checkmark are missing. (Thanks to Andrés Eduardo Caicedo, January 2018.)
Page 106
Line three: "centered z" should be "centered at z."

Page 112

Margin-figure 4.17: The figure labels the points correctly, but the caption misidentifies some of the points. (Thanks to Andrés Eduardo Caicedo, January 2018.)
Page 113
Top sidenote: The word "correspond" is misspelled. (Thanks to Andrés Eduardo Caicedo, January 2018.)
Page 114
Theorem 4.18: In the second line of the proof, "Therefore" is misspelled. (Thanks to Andrés Eduardo Caicedo, January 2018.)

Pages 116,117

Both margin-figures: A sad decimal-to-percent error occurred. The axis ticks should be labeled by 49.8%, 50%, and 50.2%. (Thanks to Samuel Wagstaff, December 2017.) Also, note that (on p.116) the 25444 split primes and 25436 inert primes is not an error -- at that point in the graph, the number of split primes (barely) exceeds the number of inert primes.

Page 117

Proof of Proposition 4.26. The q=x+yi should be a q=x+yω, near the end of the proof. (Thanks to Samuel Wagstaff, December 2017)
Page 123
Sidenote: In the phrase "there are infinitely many Gaussian prime numbers of the form x + i," the variable "x" is meant to refer to an ordinary integer (as in the exercise), not a Gaussian integer.

Part II: Modular Arithmetic


Chapter 5: The modular worlds
Page 130
The modulus does not really need to be positive, for almost everything we do. Congruence modulo a negative m is the same as congruence modulo its absolute value. In fact, one can allow congruence "mod 0" too, which is just the same as equality!
Page 138
A stray equal sign appeared, in the first centered equation. There should just be a congruence, so m is congruent to 0 mod m at the end.
Page 141
Last two paragraphs: When writing about "two roots", I mean "two distinct roots". For more on Cebotarev's Theorem, see Stevenhagen, P. and Lenstra, H. W., Jr., "Chebotarëv and his density theorem", Math. Intelligencer 18 (1996), no. 2, 26–37. (Thanks to Andrés Eduardo Caicedo, January 2018.)

Page 141

Problem 5.27: The end should read, "The solutions are x ≡ 2 mod 7 [not 3 mod 7] and x ≡ 5 mod 7." (Thanks to Samuel Wagstaff and Robert Woodley, December 2017)

Page 144

Line 3 and line 8. The subscripts (1 and 2) migrated outside the absolute value. Please put them back in, next to the letter R where they belong.
Page 145
Mid-page: "Linear polynomials play a special role." They do not "place" a special role.
Page 146
Line 8: "degree up to 19" should be "degree up to 20".
Page 149
Some accent marks are missing from the French. It should read "Premièrement, tout nombre est composé d’autant de quarrés entiers qu’il a d’unités". Note this is based on the cited edition of Tannery, and may differ from other published editions of Fermat's letters. (Thanks to Andrés Eduardo Caicedo, January 2018.)
Page 149
The cited 1828 paper of Jacobi is really an announcement, and not a sketch of the proof (as suggested in the sidenote). (Thanks to Andrés Eduardo Caicedo, January 2018.)

Page 150

Exercise 5: One should assume that m is nonzero. Perhaps even that m is positive, if you don't like negative moduli.

Chapter 6: Modular dynamics
Page 152
In the opening figure, there's a strange isolated semicolon floating around the bottom-left.

Page 154

In Problem 6.1, there are 7 equal signs that should be congruences instead.
Page 166
In sidenote 7, "precient" should be "prescient". (Thanks to Alberto Trombetta, September 2017.)

Chapter 7: Assembling the modular worlds

Chapter 8: Quadratic residues

Page 199

In the middle of the page, in the body and in the margin, I've mistakenly described the partnerships between x and -x as "(-1)-partnerships", which is incorrect. I've simply partnered each number x mod p with -x mod p. The rest of the argument (about E and O) is correct. (Thanks to Marco Schockmel, November 2017.)

Page 213

Second paragraph of the proof of Lemma 8.25: The permutation alpha sends [a,b] to <a,b] (and not the other way around). (Thanks to Marco Schockmel, November 2017.)

Page 215

Solution of Problem 8.28: 42 is not congruent to 1 modulo 7... change that 42 to 43. (Thanks to Firas Melaih, October 2017)

Part III: Quadratic forms


Chapter 9: The topograph

Page 232

Last sentence of 3rd paragraph: there should be plus/minus (\pm) signs, to read "So we prefer \pm (0,1) over \pm (0,-1)."
Page 236
Proof of Thm 9.14, end of 1st paragraph: v and w lost their over-arrows.
Page 249
Top of page, "allow us transform" should be "allow us to transform".

Chapter 10: Definite forms

Chapter 11: Indefinite forms
Page 291
In the two figures, the "v" should be in italics like the other variables.

End matter


Page 315
There should only be one entry for Sunzi (c.220-420CE).