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:
Your full name, especially if you would like acknowledgment in future editions and on this errata page.
The page and location (e.g., line 10, beginning of the Proof, etc.) of the mistake.
A brief description of the mistake, and how you might fix it (optional).
Any other relevant information, e.g., how many students worked all night on an impossible exercise because of my error.
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.
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.)
Marginnote, top-right: Zero is also typically considered a triangular number.
Marginnote, first sentence: "triangles" should be "triangle". (Thanks to Chris Shelley, November 2017.)
Paragraph below the big figure: Delete "on" in "Add only on..." (Thanks to Conner Jure, January 2018.)
Just above Prop 0.13: "are form a" should be "are from a".
Line 4: "they becomes second nature" should be "they become second nature". (Thanks to Erik Wallace, January 2018.)
Line 2: "analagous" should be "analogous". (Thanks to Alberto Trombetta, January 2018.)
The end-of-proof square comes a few lines too early in Proposition 0.27.
Last line: "mutliple" should be "multiple". (Thanks to Alberto Trombetta, January 2018.)
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.)
Exercise 26: 6N should be 6 S(N). Also "perfect squares" should be called "squares". (Thanks to Patrick McDonald, January 2018.)
Exercise 27: The parenthesis ) should be removed after "Notes". (Thanks to Chris Shelley, November 2017.)
Part I: Foundations
Chapter 1: The Euclidean algorithm
Line 13: "equaions" should be "equation".
Proposition 1.20: The integers a,b should both be nonzero.
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.)
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.)
The end-of-proof (QED) square is misplaced. It should be just above Problem 1.24.
Corollary 1.25: The integers a,b should both be nonzero here too. (Thanks to Andrés Eduardo Caicedo, December 2017.)
Proposition 1.27 and Corollary 1.29: The end-of-proof boxes are missing. (Thanks to Erik Wallace, January 2018.)
Exercise 14: The last line should read "LCM(u^2, v^2) = LCM(u,v)^2". The final "squared" was forgotten.
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.)
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
Marginnote: A primality certificate typically refers to something more than we've described: a specific theorem and relevant parameters that rapidly guarantee primality.
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.)
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.)
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.)
The exponent of 7 should be called f_7, and not f_5, in equations (2.1), (2.2), and (2.3).
Last paragraph: For the proof of existence of prime decomposition, look back to page 48. (Thanks to Erik Wallace, February 2018.)
Sidenote 19: Change "has" to "as" in "do not have 2 has a common factor". (Thanks to Jeffrey S. Haemer, January 2018.)
Line 3: change "generators" to "generates". (Thanks to Andrés Eduardo Caicedo, December 2017.)
Theorem 2.28: The proof is missing its end-of-proof box. (Thanks to Harrison Henningsen, February, 2018.)
A period wandered away from 2N, at the bottom of the page.
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.)
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.)
Exercise 1: The first "and" should be deleted. (Thanks to Jeffrey S. Haemer, January 2018.)
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.)
Exercise 17: One must require n > 1 for this construction of amicable numbers to work! (Thanks to Andrés Eduardo Caicedo, January 2017.)
Exercise 18(b): The period should go within the parenthesis at the end.
Chapter 3: Rational and constructible numbers
Marginnote above figure: Change "one associate" to "one can associate". (Thanks to Andrés Eduardo Caicedo, January 2017.)
Marginnote 3: Change "from" to "and" in "...both 1/0 from -1/0...". (Thanks to Harrison Henningsen, February, 2018.)
Marginnote 13: The "line between them" does not make sense if the circles are concentric. If you wish to play with the circles and lines, Desmos is a good free tool. Here's a Desmos graph with sliders to control the circle and see where the line goes. (Thanks to Jeffrey S. Haemer, February 2018).
An end-of-proof box is missing at the bottom of the page. (Thanks to Andrés Eduardo Caicedo, January 2018.)
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.)
Ten lines from the bottom, the x^2 + y^2 should be a u^2 + v^2. (Thanks to Andrés Eduardo Caicedo, January 2018.)
Sidenote 16: u should be a/c (not a/b) and v should be b/c. (Thanks to Jeffrey S. Haemer, February 2018.)
Second paragraph, delete "must" in "must would". (Thanks to Andrés Eduardo Caicedo, January 2018.)
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.)
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.)
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.)
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.)
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.)
I really should have mentioned Farey sequences by name, and in the historical notes.
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.)
Sidenote 41: "...on can try..." should be "...one can try..."
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)
Exercise 6(b): The variables a and b got switched. It should read "the equation ax+by = 1 implies xπ/b + yπ/a = 1/ab." Also, there's an extra period in "cosine.." (Thanks to Jeffrey S. Haemer, April 2018.)
Exercise 9: Sadly, no fraction kisses 77/133. Change 133 to 138 for a better problem. (Thanks to Jeffrey S. Haemer, April 2018.)
Exercise 11: The word "adjacent" means "tangent" here. (Thanks to Patrick McDonald, March 2018.)
Exercise 13: The number x should be irrational throughout! (Thanks to Patrick McDonald, March 2018.)
Chapter 4: Gaussian and Eisenstein integers
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.
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.)
Line three: "centered z" should be "centered at z."
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.)
Top sidenote: The word "correspond" is misspelled. (Thanks to Andrés Eduardo Caicedo, January 2018.)
Theorem 4.18: In the second line of the proof, "Therefore" is misspelled. (Thanks to Andrés Eduardo Caicedo, January 2018.)
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.
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)
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
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!
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.
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.)
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)
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.
Mid-page: "Linear polynomials play a special role." They do not "place" a special role.
Line 8: "degree up to 19" should be "degree up to 20".
Line 14: The integral should have upper-limit x, and not infinity. Its limits should match the sum. (Thanks to Adrian Shestakov, March 2018.)
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.)
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.)
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
In the opening figure, there's a strange isolated semicolon floating around the bottom-left.
In Problem 6.1, there are 7 equal signs that should be congruences instead.
In sidenote 7, "precient" should be "prescient". (Thanks to Alberto Trombetta, September 2017.)
Chapter 7: Assembling the modular worlds
Chapter 8: Quadratic residues
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.)
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.)
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
Last sentence of 3rd paragraph: there should be plus/minus (\pm) signs, to read "So we prefer \pm (0,1) over \pm (0,-1)."
Proof of Thm 9.14, end of 1st paragraph: v and w lost their over-arrows.
Top of page, "allow us transform" should be "allow us to transform".
Chapter 10: Definite forms
Chapter 11: Indefinite forms
In the two figures, the "v" should be in italics like the other variables.
There should only be one entry for Sunzi (c.220-420CE).