% List of Errata for A Friendly Introduction to Number Theory
%  by Joseph H. Silverman
% This document should be processed using plain.tex. 

\magnification=\magstep1
\def\versionnumber{2.0}
\def\versiondate{Tuesday, June 16, 1998}

\voffset=24pt
\advance\vsize by-\voffset
\footline={\ifnum\pageno>1 {\sevenbf**} \sevenrm Item has been 
corrected in the 2nd printing.\hfil
v.~\versionnumber, \versiondate\else\hfil\fi}
\headline={\ifnum\pageno>1 \tenrm Errata List for
{\tenit A Friendly Introduction to Number Theory} \hfil Page \folio
\else\hfil\fi}

% Format for corrections is  \\page number\\reference info\\
\def\\#1\\#2\\{\filbreak\smallskip\moveleft.5\parindent\hbox{\bf Page #1:\quad
#2} \par\noindent\ignorespaces}

% For corrections made in 2nd printing, use  \1 page number\\reference info\\
\def\1#1\\#2\\{\filbreak\smallskip\moveleft.5\parindent\hbox{\bf ** Page #1:\quad
#2} \par\noindent\ignorespaces}

\def\SB{should be }

% -------------------------------------------------------------------------

\centerline{Errata List for {\it A Friendly Introduction to Number
Theory}}
\centerline{by Joseph H. Silverman}
\centerline{Version \versionnumber --- \versiondate}
\bigskip
\noindent
Please send additional corrections or comments to the author at:
\smallskip
\centerline{\tt jhs@math.brown.edu}
\smallskip\noindent
or visit the {\it Friendly Introduction to Number Theory} Home Page at
\smallskip
\centerline{\tt http://www.math.brown.edu/\char"7Ejhs/frint.html}
\medskip\noindent
NOTE: Items marked with {\bf**} have been corrected in the 2nd printing.
\bigskip\noindent
The author would like to thank the following individuals for their
assistance 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.

\bigskip
% -------------------------------------------------------------------------

\\4\\Line 15\\
``nonumerical'' should be ``non-numerical''

\\9\\Line $-7$\\
``grey dots'' should read ``black dots''

\1 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.''

\\13--15\\Text\\
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.)


\1 17\\Notational Interlude\\
Replace ``number'' with ``numbers'', since ``Zahlen'' is plural.

\1 23\\List of Names\\
Langlands is Canadian.\par\noindent
Hellegouarch is French.\par\noindent
Ramikrishna is American.\par\noindent
Li is American.\par\noindent
Change ``England'' to ``United Kingdom''

\\28\\Line 5\\
Replace ``12345-417(29)=259'' by ``12345-417(29)=252''.

\1 43\\Assertion \#1\\
Replace ``$n$ can be factored'' by ``The number $n$ can be factored''

\\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.

\1 50\\Displayed Equation at Line $-4$\\
The second $+k$ should be $-k$. Thus the displayed formula should read
$$
  \left(u_0+k\cdot{m\over g}, v_0-k\cdot{a\over g}\right).
$$

\\50\\Last Paragraph\\
The argument beginning ``Further, the Linear Equation Theorem\dots'' 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 $au+mv=c$.

\1 51\\Middle of Page\\
``$943\equiv 381 \pmod{2576}$'' should be
``$943x\equiv 381 \pmod{2576}$''  (there's a missing $x$)

\\51\\Line $-5$\\
$(u,v)=(79,-29)$ should be $(u,v)=(79,29)$.

\\51\\Line $-3$\\
$(x,y)=(1106,-406)$ should be $(x,y)=(1106,406)$

\\54\\Last line of footnote\\
``discovers'' should be ``discoverers''

\\56\\Last line\\
``than it divides'' should be ``then it divides''

\1 67\\Chinese Remainder Theorem\\
``and let $a$ and $b$ be any integers'' should be
``and let $b$ and $c$ be any integers'' 

\\75\\Middle of page\\
``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''

\1 81\\Line $-3$\\
``$2^2-1-3$'' should be ``$2^2-1=3$''.

\\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''.
\par\noindent
Add the new Mersenne prime with $p=1257787$ discovered by Slowinski 
and Gage in 1996.
\par\noindent
Add the new Mersenne primes:
\par
$p=1398269$, discovered by Armengaud (1996).
\par
$p=2976221$, discovered by Spence (1997).
\par
$p=3021377$, discovered by Clarkson (1998).
\par\noindent
These were found as part of Woltman's ``Great Internet Mersenne Prime Search''
(GIMPS). Include the internet address for those wishing to participate:
\par
{\tt http://www.mersenne.org/prime.htm}
\par\noindent
Include the following internet address for those who want further 
(up-to-date) information about Mersenne primes:
\par
{\tt http://www.utm.edu/research/primes/mersenne.shtml}

\1 86\\Line $-11$\\
Add a question mark at the end of the sentence ``\dots or
are there other perfect numbers.''

\1 88\\Line 15\\
``$\sigma(mn)=\sigma(n)\sigma(n)$'' should be
``$\sigma(mn)=\sigma(m)\sigma(n)$''

\\93\\Line 17\\
``Further, since we alway reduce mod-'' should be
``Further, since we always reduce mod-'' (add an ``s'' to ``alway'').

\1 97\\Displayed Formulas\\
The two displayed formulas should be centered, not right
justified.

\\104\\Lines $-4$ and $-1$\\
The encryption of the number 30302512 should be 118084566, not
23054767. Thus line $-4$ should read
$$
  30302512^{79921} \equiv 23054767 \pmod{163276871},
$$
and the bottom line on the page should read
$$
  149419241,\qquad 62721998,\qquad 118084566,\qquad 40481382.
$$

\\104\\Remark\\
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.

\\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).

\1 107\\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.

\\110\\Verification\\
The way the products $d_id_j$ are written out is rather confusing.
Throughout this verification it would be better to write 
$$
  d_1e_1, d_1e_2, \dots, d_1e_s, d_2e_1, d_2e_2,
    \dots, d_2e_s, \dots , d_re_1, d_re_2, \dots , d_re_s.
$$

\1 114\\Exercise 20.2(e)\\
Change ``where $p$ is a prime'' to ``where $p$ is an odd prime''

\1 119\\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).

\1 120\\1\\
Change ``Riemann hypothesis'' to ``generalized Riemann hypothesis''

\\136\\Line 10\\
Change ``an even power of $p$'' to ``an even power of $g$''.

\\142\\Lines 6--8\\
These eight congruences should be modulo~17, not modulo~13.

\1 149\\Lines $-7$ and $-5$\\
``the Langlands' Program'' should be ``the Langlands Program'' (twice)

\1 150\\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.)

\1 160\\Line 9\\
Change ``following table'' to ``table on the following page''

\1 160 and 161\\Second Column of Both Table\\
The headings should read ``$p$ is any 
prime ${}\equiv1~(\hbox{mod}~4)$'', not ``mod~$p$''.

\\162\\Display in middle of page\\
In the display with $M=\cdots<p$, the last fraction should be $2p-2\over p$,
rather than $p-2\over p$. Thus the entire line should read
$$
  M={A^2+B^2\over p}\le{(p-1)^2+1^2\over p}
  =p-{2p-2\over p} < p.
$$

\1 162\\Display in middle of page\\
$\displaystyle -{1\over2}\le u,v\le {1\over2}M$
should be
$\displaystyle -{1\over2}M\le u,v\le {1\over2}M$.
(The first $M$ is missing.)

\\166\\Second display from bottom\\
The equation $18=3^3+9^2$ should be $18=3^2+3^2$. (Notice there are two 
errors, the exponent on the first ``3'' should be ``2'', and the ``9'' should
be a ``3''.)

\1 185\\Line $-7$\\
``There in no known'' should be ``There is no known''

\1 196\\Line 4\\
``since any solution will to Pell's equation satisfy'' should be
``since any solution to Pell's equation will satisfy''

\1 207\\Footnote\\
Change ``it's'' to ``its''

\\210\\Lines 10 and 11\\
The sentence ``This line has slope $19/6$ and is given by the
equation $y=19x/6+28/3$'' should read
``This line has slope $-19/6$ and is given by the
equation $y=-19x/6-10/3$''
(Need to add two minus signs and change the $y$-intercept of
the line.)

\1 212\\Exercise 32.2(d)\\
This should read ``Let $S$ be the point you found in (b),\dots'',
not ``\dots in (c)\dots''.

\1 214\\Tables 1 and 2\\
In both tables, the reference should be to $E_1$, not to $E_2$.

\1 219\\Torsion Theorem\\
The statement of the theorem should include the assumption that the
discriminant $\Delta(E)$ is not zero.

\1 220\\Siegel's Theorem\\
The statement of the theorem should include the assumption that the
discriminant $\Delta(E)$ is not zero.

\1 231\\Line $-5$\\
``throught'' should be ``through''

\\233\\Exercise 35.3(a)\\
The ``?'' at the end of the sentence should be a ``.''.

\1 241\\Top two paragraphs\\
Ribet actually proved his result during the summer of 1986
and gave a series of lectures on it during the Fall, although
his paper didn't appear in print until 1990. Thus the six
(or really seven) years following Ribet's proof that Wiles spent
working on the Modularity Conjecture were Summer 1986 to Spring 1993.
Wiles' completed paper appeared in 1995.

\1 248\\Exercise A4.1(e)\\
``primtive'' should be ``primitive''

\1 248\\Exercise A5.2, Possibility (i)\\
The description of the terminating value and length of the algorithm
are not correct. Replace the second and third sentences ``In this case 
\dots {\it length of the algorithm}'' with the following:
\par
In this case we say that the algorithm {\it terminates} at the last
non-repeated value, and the number of distinct entries in the list 
is called the {\it length of the algorithm}.

\1 250\\Exercise 10.2\\
``it true for every number $a$'' should be 
``is true for every number $a$''

\1 254\\Exercise A25.2 (i) and (ii)\\
``satifies'' should be ``satisfies'' (twice)

\\257\\New Exercise A.33.3\\
(a) Show that the equation $y^2=x^3+x^2$ has infinitely many 
solutions in integers $x,y$. (Hint. Try substituting $y=tx$.)
\par\noindent
(b) Does your answer in (a) mean that Siegel's Theorem is incorrect? 
Explain.
\par\noindent
(c) Show that the equation $y^2=x^3-x^2-x+1$ has infinitely many 
solutions in integers $x,y$.

\1 258\\Exercise A34.1(e), Lines 2--3\\
``satisfying that $4p=A^2+3B^2$'' should be 
``satisfying  $4p=A^2+3B^2$''

\1 258\\Exercise A36.1, Line 2\\
``listed in exercise 48'' should be ``listed in exercise A34.1''


\bye
