Errata for A FRIENDLY INTRODUCTION TO NUMBER THEORY by Joseph H. Silverman Version 2.0 - Tuesday, June 16, 1998 Please send additional corrections or comments to me at jhs@math.brown.edu or visit the "Friendly Introduction to Number Theory" Home Page at http://www.math.brown.edu/~jhs/frint.html Thanks to the following people for help in compiling this errata sheet: Arthur Baragar Nigel Boston David Boyd Seth Braver Michael Catalano-Johnson L. Chang Robin Chapman Miguel Cordero John Cremona Lisa Fastenberg Nicholas Fiori Fumiyasu Funami Jim Funderburk Rob Gross Tom Hagedorn Ron Jacobowitz Jerry S. Kelly, Hershy Kisilevsky Hendrik Lenstra Gordon S. Lessells Ken Levasseur Nidia Lopez J. Metzger Rachel Pries Ken Ribet David Rohrlich Alfred Tang Wenchao Zhou %%%%%%%%%%%%%%%%%%%%%%%%%%% ERRATA %%%%%%%%%%%%%%%%%%%%%%%%%%% *** ITEMS MARKED WITH "***" HAVE BEEN CORRECTED IN THE 2ND PRINTING Page 4, Line 15 "nonumerical" should be "non-numerical" Page 9, Line -7 "grey dots" should read "black dots". *** Page 10, Exercise 1.1 The first square-triangle number is 1, not 36. Replace the first sentence with: "The first two numbers which are both squares and triangles are 1 and 36." Pages 13-15 The book makes use of the notion of numbers having "no common factors" in an informal way, without having given a formal definition. This allows the class to see some substantial and interesting number theory very early in the course, but could cause some difficulty for the students. The teacher will want to say a few words about what it means to have common factors, and it might be good if there were a brief comment in the text together with an assurance that such matters will be discussed more carefully in Chapter 5. (This might also be mentioned in the Guide for the Instructor on pages 3-4.) *** Page 17, Notational Interlude Replace "number" with "numbers", since "Zahlen" is plural. *** Page 23, List of Names Langlands is Canadian. Hellegouarch is French. Ramikrishna is American. Li is American. Change "England" to "United Kingdom" Page 28, Line 5 Replace "12345-417(29)=259" by "12345-417(29)=252". *** Page 43, Assertion #1 Replace "n can be factored" by "The number n can be factored" Page 45, Line 5 "As a practical matter, how can we write it as a product of primes." should end with a question mark, not a period. *** Page 50, Displayed Equation at Line -4 The second +k should be a -k. Page 50, Last Paragraph The argument beginning "Further, the Linear Equation Theorem..." is both inaccurate and confusing, although the answer given is correct. If we take all of the solutions to au+mv=g, then (1) the natural thing to do is multiply them by c/g to get solutions to au+mv=c, but that is not what is done; and (2) it's not clear that we're getting all of the solutions to au+mv=c. *** Page 51, Middle of Page 943 = 381 (mod 2576) is missing an "x", it should read 943x = 381 (mod 2576) Page 51, Line -5 (u,v)=(79,-29) should be (u,v)=(79,29). Page 51, Line -3 (x,y)=(1106,-406) should be (x,y)=(1106,406). Page 54, Last line of footnote "discovers" should be "discoverers" Page 56, Last line "than it divides" should be "then it divides" *** Page 67, Chinese Remainder Theorem "and let $a$ and $b$ be any integers" should be "and let $b$ and $c$ be any integers" Page 75, Middle "This function counts how many even numbers there are less than x" should be "This function counts how many even numbers there are less than or equal to x" *** Page 81, Line -3 "2^2-1-3" should be "2^2-1=3" Page 82, Table Put a little more space after the commas following "521" and "1279", since otherwise it looks like the first entry is the six digit number "521607". Add the new Mersenne prime with p=1257787, discovered by Slowinski and Gage in 1996. Add the new Mersenne primes p=1398269, discovered by Armengaud (1996); p=2976221, discovered by Spence (1997); p=3021377, discovered by Clarkson (1998). These were found as part of Woltman's "Great Internet Mersenne Prime Search" (GIMPS). Include the internet address for those wishing to participate: http://www.mersenne.org/prime.htm Include the following internet address for those who want further (up-to-date) information about Mersenne primes: http://www.utm.edu/research/primes/mersenne.shtml *** Page 86, Line -11 Add a question mark at the end of the sentence "...or are there other perfect numbers." *** Page 88, Line 15 "\sigma(mn)=\sigma(n)\sigma(n)" should be "\sigma(mn)=\sigma(m)\sigma(n)" Page 93, Line 17 "Further, since we alway reduce mod-" should be "Further, since we always reduce mod-" (add an "s" to "always") *** Page 97, Displayed Formulas The two displayed formulas should be centered, not right justified. Page 104, Lines -4 and -1 The encryption of the number 30302512 should be 118084566, not 23054767. Page 104 Put in a remark (or an exercise) saying that the method still works even if gcd(x,m) > 1, where x is the message. This is true because m is square-free, as explained in Exercise A17.2. Page 107, End of Chapter 18 Possibly include a brief discussion about what happens if the plaintext a is not relatively prime to m=pq. Point out that this is extremely unlikely to occur, and that even if it does occur, Exercise A17.2(a) shows that decoding still works (because m is square-free). *** Page 107, Exercise 18.3 Mention that the data for this exercise is available on the FRINT web site for those who don't wish to retype it. Page 110, Verifiction The way that the products d_id_j are written out is rather confusing. It would be better to write d_1e_1,d_1e_2,...,d_1e_s,d_2e_1,d_2e_2,...,d_2e_s,..., d_re_1,d_re_2,...,d_re_s. *** Page 114, Exercise 20.2(e) Change "where p is a prime" to "where p is an odd prime" *** Page 119, Line -7 The sentence "Artin's original conjecture was later generalized to numbers other than 2" is inaccurate. Artin actually made his conjecture for any nonsquare integer different from -1 (see pages viii-x of Artin's Collected Papers). *** Page 120, Line 1 Change "Riemann hypothesis" to "generalized Riemann hypothesis" Page 136, Line 10 Change "an even power of p" to "an even power of g" Page 142, Lines 6-8 These eight congruences should be modulo 17, not modulo 13. *** Page 149, Lines -7 and -5 "the Langlands' Program" should be "the Langlands Program" (twice) *** Page 150, Line 1 Gauss' first name is spelled as both Karl and Carl. Change both to Karl, which is consistant with usage in the rest of the book. (It has been noted that this seems to be the less preferred usage.) *** Page 160, Line 9 Change "following table" to "table on the following page" *** Pages 160 and 161, Second Column of Both Tables The headings should read "p is any prime = 1 (mod 4)", not "mod p". Page 162, Display in middle of page In the display with "M=...