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\OT1/cmr/m/n/10 [Acknowledgments: A state-ment thank-ing those in-di-vid-u-als 
who have helped
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\OT1/cmr/m/it/10 Dyer Con-jec-ture \OT1/cmr/m/n/10 which gives a sim-ple con-je
c-tural cri-te-rion to de-cide whether
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\OT1/cmr/m/n/10 They are de-noted by $\U/msb/m/n/10 Z$ \OT1/cmr/m/n/10 be-cause
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[]\OT1/cmr/m/n/10 1881988129206079638386972394616504398071635633794173827007
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[][] 
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[]\OT1/cmtt/m/n/10 ? for(n=1,10,print(n*1000,"\t",pi(n*1000),"\t",n*1000/(log(n
*1000)-1)))[] 
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[]\OT1/cmtt/m/n/10 http://numbers.computation.free.fr/Constants/Primes/Pix/pixp
roject.html[] 
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[]      \OT1/cmtt/m/n/10 Li(10^22)     = 201467286691248261498.1505...    (usin
g Maple)[] 
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\U/msb/m/n/10 Z\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 3\U/msb/m/n/10 Z \OT1/cmr/m/n/
10 = \OMS/cmsy/m/n/10 ff[] \OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/
10 3\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 3\OM
L/cmm/m/it/10 ; []\OMS/cmsy/m/n/10 g\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 f[] \OM
L/cmm/m/it/10 ; \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ; \OT1/cm
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10 ^^@\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ; \
OT1/cmr/m/n/10 5\OML/cmm/m/it/10 ; []\OMS/cmsy/m/n/10 gg \OT1/cmr/m/n/10 = \OMS
/cmsy/m/n/10 f\OT1/cmr/m/n/10 [0]\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 [1]\OML/cmm
/m/it/10 ; \OT1/cmr/m/n/10 [2]\OMS/cmsy/m/n/10 g
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\OT1/cmr/bx/n/10 Definition 1.24 (Com-plete Set of Residues). \OT1/cmr/m/n/10 A
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[] 
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[]\OT1/cmtt/m/n/10 bezout(x,y): gives a 3-dimensional row vector [u,v,d] such t
hat[] 
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[]\OT1/cmtt/m/n/10 ? x = 094860934869840698340958093485903450983409580934850983
4905809345[] 
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[]\OT1/cmtt/m/n/10 %36 = 948609348698406983409580934859034509834095809348509834
905809345[] 
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[]\OT1/cmtt/m/n/10 %37 = 948609348698406983409580934859034509834095809348509834
905809399[] 
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[]\OT1/cmr/m/n/10 What was the most re-cent prime year? I.e., which of $[] \OML
/cmm/m/it/10 ; \OT1/cmr/m/n/10 2002\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 2001\OML/
cmm/m/it/10 ; \OT1/cmr/m/n/10 2000\OML/cmm/m/it/10 ; []$
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[]\OT1/cmr/m/n/10 Meanwhile, hack-ers in The Col-lec-tive see both $\OML/cmm/m/
it/10 g[]\OT1/cmr/m/n/10 ([]\OML/cmm/m/it/10 p\OT1/cmr/m/n/10 )$ and $\OML/cmm/
m/it/10 g[]\OT1/cmr/m/n/10 ([]\OML/cmm/m/it/10 p\OT1/cmr/m/n/10 )$,
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[] \OT1/cmtt/m/n/10 IEEE Trans. Information Theory IT-22 (1976), no. 6, 644--65
4.[] 
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[]\OT1/cmtt/m/n/10 ? dislog(x,g, s) = s=g; for(n=1,znorder(g),if(x==s, return(n
), s=s*g)); 0;[] 
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[]\OT1/cmtt/m/n/10 ? znlog(a,g)        \\ builtin super-optimized version takes
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[]\OT1/cmtt/m/n/10 We are pleased to announce a new record for the discrete log
arithm[] 
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[]\OT1/cmtt/m/n/10 quadri-processors Digital Alpha Server 8400 computer. The ap
proach[] 
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[]\OT1/cmtt/m/n/10 that we followed is a careful implementation of the general 
Function[] 
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[]\OT1/cmtt/m/n/10 Field Sieve as described from a theoretical point of view by
 Adleman[] 
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[]\OT1/cmtt/m/n/10 As far as we know, the largest such computation previously d
one was[] 
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[]\OT1/cmtt/m/n/10 performed in GF(2^401) [GoMc92] using an algorithm due to Co
ppersmith[] 
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[]\OT1/cmtt/m/n/10 logarithms in GF(2^521) on a 525 MHz quadri-processor alpha 
server[] 
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[]\OT1/cmtt/m/n/10 8400 computer is approximatively 12 hours for each, once the
 sieving[] 
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[]\OT1/cmtt/m/n/10 step (21 days) and the linear algebra step (10 days) is perf
ormed.[] 
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[]\OT1/cmtt/m/n/10 Antoine JOUX    (DCSSI, Issy les Moulineaux, France, Antoine
.Joux@ens.fr),[] 
 []

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[]\OT1/cmtt/m/n/10 ? nextgoodprime(p) = while(!isprime((p-1)\2), p=nextprime(p+
1)); p[] 
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[]\OT1/cmtt/m/n/10 %26 = Mod(17037287637415625385373411504, 9345098309485093845
0983409611)[] 
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[]\OT1/cmtt/m/n/10 %27 = Mod(2201425894324369970772940547, 93450983094850938450
983409611)[] 
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[]\OT1/cmtt/m/n/10 %28 = Mod(25591938014843312529239952955, 9345098309485093845
0983409611)[] 
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[]\OT1/cmtt/m/n/10 %29 = Mod(25591938014843312529239952955, 9345098309485093845
0983409611)[] 
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\OT1/cmr/m/n/10 Encode your mes-sage as a se-quence of num-bers mod-ulo $\OML/c
mm/m/it/10 n$ \OT1/cmr/m/n/10 (see Sec-tion 2.2.2[]): 
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[]  \OMS/cmsy/m/n/10 $  \OT1/cmr/m/n/10 8 + 27 \OMS/cmsy/m/n/10 ^^A \OT1/cmr/m/
n/10 1 + 27[] \OMS/cmsy/m/n/10 ^^A \OT1/cmr/m/n/10 18 + 27[] \OMS/cmsy/m/n/10 ^
^A \OT1/cmr/m/n/10 22 + 27[] \OMS/cmsy/m/n/10 ^^A \OT1/cmr/m/n/10 1 + 27[] \OMS
/cmsy/m/n/10 ^^A \OT1/cmr/m/n/10 18 + 27[] \OMS/cmsy/m/n/10 ^^A \OT1/cmr/m/n/10
 4 = 1808939906
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\OML/cmm/m/it/10 '\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 ) = \OML/c
mm/m/it/10 '\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 p \OMS/cmsy/m/n/10 ^^A \OML/cmm/m
/it/10 q\OT1/cmr/m/n/10 ) = \OML/cmm/m/it/10 '\OT1/cmr/m/n/10 (\OML/cmm/m/it/10
 p\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^A \OML/cmm/m/it/10 '\OT1/cmr/m/n/10 (\OM
L/cmm/m/it/10 q\OT1/cmr/m/n/10 ) = (\OML/cmm/m/it/10 p \OMS/cmsy/m/n/10 ^^@ \OT
1/cmr/m/n/10 1)(\OML/cmm/m/it/10 q \OMS/cmsy/m/n/10 ^^@ \OT1/cmr/m/n/10 1) = \O
ML/cmm/m/it/10 pq \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 p \OMS/cmsy/m/n/10 ^^@ 
\OML/cmm/m/it/10 q \OT1/cmr/m/n/10 + 1 = 323 \OMS/cmsy/m/n/10 ^^@ \OT1/cmr/m/n/
10 17 \OMS/cmsy/m/n/10 ^^@ \OT1/cmr/m/n/10 19 + 1 = 288\OML/cmm/m/it/10 :
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[]\OT1/cmtt/m/n/10 {alphabet=[" ","A","B","C","D","E","F","G","H","I","J","K","
L","M",[] 
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[]           \OT1/cmtt/m/n/10 "N","O","P","Q","R","S","T","U","V","W","X","Y","
Z"];[] 
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[]     \OT1/cmtt/m/n/10 for(i=1,length(w), n = n + 27^(i-1)*letter_to_number(w[
i]));[] 
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[]\OT1/cmtt/m/n/10 encrypt(message, n, e) = lift(Mod(message_to_number(message)
,n)^e);[] 
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[]\OT1/cmtt/m/n/10 decrypt(secret, n, d)  = number_to_message(lift(Mod(secret,n
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[]\OT1/cmtt/m/n/10 %22 = 611165840845056469708563420184597685050990858094998688
9525704...[] 
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[]\OT1/cmtt/m/n/10 ? \p150        \\ set precision of floating-point computatio
ns.[] 
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Overfull \hbox (11.99707pt too wide) in paragraph at lines 698--698
[]   \OT1/cmtt/m/n/10 realprecision = 154 significant digits (150 digits displa
yed)[] 
 []


Overfull \hbox (27.74693pt too wide) in paragraph at lines 698--698
[]\OT1/cmtt/m/n/10 %30 = 48.000000000000000000000000000000000000000000000000000
000....[] 
 []

[45]
Overfull \hbox (32.99689pt too wide) in paragraph at lines 784--784
[]\OT1/cmtt/m/n/10 ? \r rsa   \\ load the file rsa.gp, available at Lecture 9 w
eb page.[] 
 []


Overfull \hbox (11.99707pt too wide) in paragraph at lines 784--784
[]\OT1/cmtt/m/n/10 ? for(a=2,20, if(Mod(a,n)^m!=1,print(a)))   \\ prints nothin
g...[] 
 []

[46]
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[]  \OT1/cmtt/m/n/10 http://www.rsasecurity.com/rsalabs/challenges/factoring/nu
mbers.html[] 
 []


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[]\OT1/cmtt/m/n/10 188198812920607963838697239461650439807163563379417382700763
356422988859[] 
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[]\OT1/cmtt/m/n/10 715234665485319060606504743045317388011303396716199692321205
734031879550[] 
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]
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[51] [52] [53] [54]
Overfull \hbox (6.84212pt too wide) in paragraph at lines 300--302
\OT1/cmr/bx/n/10 Corollary 3.12. \OT1/cmr/m/it/10 If $\OML/cmm/m/it/10 x[] \OMS
/cmsy/m/n/10 ^^Q \OML/cmm/m/it/10 a\OT1/cmr/m/n/10 ([]\OML/cmm/m/it/10 p\OT1/cm
r/m/n/10 )$ \OT1/cmr/m/it/10 has no so-lu-tions if and only if $\OML/cmm/m/it/1
0 a[] \OMS/cmsy/m/n/10 ^^Q
 []


Overfull \hbox (16.97096pt too wide) detected at line 317
\OT1/cmr/m/n/10 1[] = 1\OML/cmm/m/it/10 ;  \OT1/cmr/m/n/10 2[] = 4\OML/cmm/m/it
/10 ;  \OT1/cmr/m/n/10 3[] = 9\OML/cmm/m/it/10 ;  \OT1/cmr/m/n/10 4[] = 5\OML/c
mm/m/it/10 ;  \OT1/cmr/m/n/10 5[] = 3\OML/cmm/m/it/10 ;  \OT1/cmr/m/n/10 6[] = 
3\OML/cmm/m/it/10 ;  \OT1/cmr/m/n/10 7[] = 5\OML/cmm/m/it/10 ;  \OT1/cmr/m/n/10
 8[] = 9\OML/cmm/m/it/10 ;  \OT1/cmr/m/n/10 9[] = 4\OML/cmm/m/it/10 ;  \OT1/cmr
/m/n/10 10[] = 1\OML/cmm/m/it/10 :
 []


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\OT1/cmr/m/n/10 1[] = 1\OML/cmm/m/it/10 ;  \OT1/cmr/m/n/10 2[] = \OMS/cmsy/m/n/
10 ^^@\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ;  \OT1/cmr/m/n/10 3[] = 1\OML/cmm/m/it
/10 ;  \OT1/cmr/m/n/10 4[] = 1\OML/cmm/m/it/10 ;  \OT1/cmr/m/n/10 5[] = 1\OML/c
mm/m/it/10 ;  \OT1/cmr/m/n/10 6[] = \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1\OML/c
mm/m/it/10 ;  \OT1/cmr/m/n/10 7[] = \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1\OML/c
mm/m/it/10 ;  \OT1/cmr/m/n/10 8[] = \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1\OML/c
mm/m/it/10 ;  \OT1/cmr/m/n/10 9[] = 1\OML/cmm/m/it/10 ;  \OT1/cmr/m/n/10 10[] =
 \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :
 []


Overfull \hbox (8.46613pt too wide) in paragraph at lines 331--335
\OT1/cmr/m/it/10 Example \OT1/cmr/m/n/10 3.14\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10
 De-ter-mine whether or not $3$ is a square mod-ulo $\OML/cmm/m/it/10 p \OT1/cm
r/m/n/10 = 726377359$. 
 []


Overfull \hbox (17.24702pt too wide) in paragraph at lines 338--338
[]\OT1/cmtt/m/n/10 %5 = Mod(726377358, 726377359)   \\ class of -1 modulo 72637
7359.[] 
 []

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Overfull \hbox (8.35721pt too wide) detected at line 493
\OT1/cmr/m/n/10 (1\OML/cmm/m/it/10 a\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^A \OT1
/cmr/m/n/10 (2\OML/cmm/m/it/10 a\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^A \OT1/cmr
/m/n/10 (3\OML/cmm/m/it/10 a\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^A [] ^^A [] ^^
Q \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 "[] \OMS/cmsy/m/n/10 ^^A \OT1/cmr/m/n/10 1)
 \OMS/cmsy/m/n/10 ^^A \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 "[] \OMS/cmsy/m/n/10 ^^
A \OT1/cmr/m/n/10 2) [] [] ([]\OML/cmm/m/it/10 p\OT1/cmr/m/n/10 )\OML/cmm/m/it/
10 ;
 []

[58]
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\OML/cmm/m/it/10 K \OT1/cmr/m/n/10 = [] \OML/cmm/m/it/10 :
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\openout2 = `chapters/contfrac.aux'.

 (chapters/contfrac.tex
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[] 
 []


Overfull \hbox (0.72789pt too wide) in paragraph at lines 73--76
\OT1/cmr/m/it/10 Exercise \OT1/cmr/m/n/10 4.3\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10
 Com-pute $\OML/cmm/m/it/10 p[]$ \OT1/cmr/m/n/10 and $\OML/cmm/m/it/10 q[]$ \OT
1/cmr/m/n/10 for the con-tin-ued frac-tions $[\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/
10 3\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OM
L/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m
/it/10 ; \OT1/cmr/m/n/10 3]$
 []

[64] [65] [66] [67]
Overfull \hbox (31.9004pt too wide) detected at line 259
\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 = [\OML/cmm/m/it/10 a[]; a[]; [] ; a[]; []\O
T1/cmr/m/n/10 ] = [\OML/cmm/m/it/10 a[]; a[]; [] ; a[]; a[] \OT1/cmr/m/n/10 + \
OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 ] = [\OML/cmm/m/it/10 a[]; a[]; [] ; a[]; a[
]; []\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 :
 []

[68]
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[]    \OT1/cmtt/m/n/10 pp=1; qq=0; p=v[1]; q=1;    \\ pp is p_{n-1} and p is p_
n.[] 
 []

[69]
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[]\OT1/cmtt/m/n/10 %38 = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2
, 2, ...][] 
 []


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[]\OT1/cmtt/m/n/10 %39 = [3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215
, ...][] 
 []


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[]\OT1/cmtt/m/n/10 ? [c[1]*1.0, c[3]*1.0, c[5]*1.0, c[7]*1.0]  \\ odd ones conv
erge up to pi[] 
 []


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[]\OT1/cmtt/m/n/10 ? [c[2]*1.0,c[4]*1.0,c[6]*1.0,c[8]*1.0]     \\ even ones swo
op down on pi.[] 
 []

[70]
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\OT1/cmr/bx/n/10 Proposition 4.16. \OT1/cmr/m/it/10 If $\OML/cmm/m/it/10 x$ \OT
1/cmr/m/it/10 is a ra-tio-nal num-ber then the se-quence $\OML/cmm/m/it/10 a[];
 a[]; a[]; []$
 []

[71] [72]
Overfull \hbox (4.53398pt too wide) in paragraph at lines 514--521
\OT1/cmr/m/n/10 can be ex-pressed as a poly-no-mial in $\OML/cmm/m/it/10 ^^K$ \
OT1/cmr/m/n/10 with ra-tio-nal co-ef-fi-cients, so $[\OML/cmm/m/it/10 a[]; a[];
 []\OT1/cmr/m/n/10 ]$
 []

[73]
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[]\OT1/cmtt/m/n/10 %5 = [1, 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 1
2, 2, 3,[] 
 []


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[] \OT1/cmtt/m/n/10 2, 1, 3, 4, 1, 1, 2, 14, 3, 12, 1, 15, 3, 1, 4, 534, 1, 1, 
5, 1, 1,[] 
 []


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[] \OT1/cmtt/m/n/10 121, 1, 2, 2, 4, 10, 3, 2, 2, 41, 1, 1, 1, 3, 7, 2, 2, 9, 4
, 1, 3, 7,[] 
 []


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[] \OT1/cmtt/m/n/10 6, 1, 1, 2, 2, 9, 3, 1, 1, 69, 4, 4, 5, 12, 1, 1, 5, 15, 1,
 4, 1, 1,[] 
 []

[74]
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\OT1/cmr/m/n/10 sim-ple con-tin-ued frac-tion has un-bounded par-tial quo-tient
s? Does
 []

[75]
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[]\OT1/cmtt/m/n/10 %16 = [2, 2, 7, 2, 1, 5, 1, 1, 1, 1, 1, 1, 328210621945, 2, 
1, 1, 1, 1, 7][] 
 []


Overfull \hbox (69.74657pt too wide) in paragraph at lines 710--710
[]\OT1/cmtt/m/n/10 %18 = [2, 2, 7, 2, 1, 5, 1, 1, 1, 1, 1, 1, 18850095183555629
36415046, 1, 4][] 
 []


Overfull \hbox (69.74657pt too wide) in paragraph at lines 710--710
[]\OT1/cmtt/m/n/10 ? x*1.0  \\ notice that no repeat is immediately evident in 
the digits of x[] 
 []


Overfull \hbox (90.74638pt too wide) in paragraph at lines 710--710
[]\OT1/cmtt/m/n/10 %19 = 2.4681570054587990642058747075643358461138549519105796
72472056147647517..[] 
 []


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[]\OT1/cmtt/m/n/10 ? \\ in fact, the length of the period of the decimal expans
ion[] 
 []

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\OT1/cmr/bx/n/10 Proposition 4.24. \OT1/cmr/m/it/10 The units of $\U/msb/m/n/10
 Z\OT1/cmr/m/n/10 [[]]$ \OT1/cmr/m/it/10 are ex-actly the el-e-ments of norm $\
OMS/cmsy/m/n/10 ^^F\OT1/cmr/m/n/10 1$
 []

[77] [78] [79]
Overfull \hbox (21.42859pt too wide) in paragraph at lines 950--952
[]\OT1/cmr/m/n/10 If $\OML/cmm/m/it/10 c[] \OT1/cmr/m/n/10 = \OML/cmm/m/it/10 p
[]=q[]$ \OT1/cmr/m/n/10 is the $\OML/cmm/m/it/10 n$\OT1/cmr/m/n/10 th con-ver-g
ent of the con-tin-ued frac-tion $[\OML/cmm/m/it/10 a[]; a[]; [] ; a[]\OT1/cmr/
m/n/10 ]$
 []

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Overfull \hbox (20.56493pt too wide) detected at line 303
\OML/cmm/m/it/10 f\OMS/cmsy/m/n/10 j[] \OT1/cmr/m/n/10 = 458(4\OML/cmm/m/it/10 
x \OMS/cmsy/m/n/10 ^^@ \OT1/cmr/m/n/10 3\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 )[] +
 214(4\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^@ \OT1/cmr/m/n/10 3\OML/cmm/m/it/10
 y\OT1/cmr/m/n/10 )(\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 17\OML/cmm/m/it/10 x \O
T1/cmr/m/n/10 + 13\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 ) + 25(\OMS/cmsy/m/n/10 ^^@
\OT1/cmr/m/n/10 17\OML/cmm/m/it/10 x \OT1/cmr/m/n/10 + 13\OML/cmm/m/it/10 y\OT1
/cmr/m/n/10 )[] = [] = \OML/cmm/m/it/10 x[] \OT1/cmr/m/n/10 + \OML/cmm/m/it/10 
y[]\OT1/cmr/m/n/10 !!
 []

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(/usr/share/texmf/tex/latex/amsfonts/ueuf.fd
File: ueuf.fd 1995/01/05 v2.2e AMS font definitions
) [91] [92]
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[]\OT1/cmtt/m/n/10 \\ and a matrix that transforms the input form to the reduce
d form.[] 
 []


Overfull \hbox (53.9967pt too wide) in paragraph at lines 661--661
[]   \OT1/cmtt/m/n/10 if(D>=0 || a<0, error("reduce: (a,b,c) must be positive d
efinite."));[] 
 []


Overfull \hbox (17.24702pt too wide) in paragraph at lines 661--661
[]         \OT1/cmtt/m/n/10 print([a,b,c], "  \t(1)"),  \\ backslash t means ``
tab''[] 
 []

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Overfull \hbox (9.62663pt too wide) in paragraph at lines 730--732
\OT1/cmr/bx/n/10 Definition 5.20. \OT1/cmr/m/n/10 A bi-nary quadratic form $(\O
ML/cmm/m/it/10 a; b; c\OT1/cmr/m/n/10 )$ is \OT1/cmr/m/it/10 prim-i-tive \OT1/c
mr/m/n/10 if $[](\OML/cmm/m/it/10 a; b; c\OT1/cmr/m/n/10 ) =
 []


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Overfull \hbox (53.9967pt too wide) in paragraph at lines 1026--1026
[]   \OT1/cmtt/m/n/10 if(D>=0 || a<0, error("reduce: (a,b,c) must be positive d
efinite."));[] 
 []


Overfull \hbox (27.74693pt too wide) in paragraph at lines 1026--1026
[]   \OT1/cmtt/m/n/10 if (D > 0 || D%4 == 2 || D%4==3, error("Invalid discrimin
ant"));[] 
 []


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[]   \OT1/cmtt/m/n/10 return(eval(forms));    \\ eval gets rid of the annoying 
quotes.[] 
 []


Overfull \hbox (38.24684pt too wide) in paragraph at lines 1071--1071
[]\OT1/cmtt/m/n/10 %23 = [[1, 1, 105], [3, -1, 35], [3, 1, 35], [5, -1, 21], [5
, 1, 21],[] 
 []

[98]
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\OT1/cmr/bx/n/10 Theorem 5.22 (Heeg-ner, Stark-Baker, Gold-feld-Gross-Zagier). 
\OT1/cmr/m/it/10 Sup-
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LaTeX Font Info:    Font shape `OMS/cmr/m/it' in size <10> not available
(Font)              Font shape `OMS/cmsy/m/n' tried instead on input line 1077.


Overfull \hbox (15.77776pt too wide) in paragraph at lines 1080--1082
[]$\OML/cmm/m/it/10 h[] \OT1/cmr/m/n/10 = 3$ \OT1/cmr/m/it/10 only for $\OML/cm
m/m/it/10 D \OT1/cmr/m/n/10 = \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 23\OML/cmm/m/
it/10 ; \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 31\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/
10 ^^@\OT1/cmr/m/n/10 59\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 
83\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 107\OML/cmm/m/it/10 ; 
\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 139\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 ^^@\
OT1/cmr/m/n/10 211\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 283\OM
L/cmm/m/it/10 ; \OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 307\OML/cmm/m/it/10 ;$ 
 []


Overfull \hbox (0.8307pt too wide) in paragraph at lines 1104--1108
\OT1/cmr/bx/n/10 Definition 5.23. \OT1/cmr/m/n/10 Let $\OML/cmm/m/it/10 f[] \OT
1/cmr/m/n/10 = (\OML/cmm/m/it/10 a[]; b[]; c[]\OT1/cmr/m/n/10 )$ and $\OML/cmm/
m/it/10 f[] \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 a[]; b[]; c[]\OT1/cmr/m/n/10 )$
 be two quadratic
 []


Overfull \hbox (25.63708pt too wide) in paragraph at lines 1110--1115
\OT1/cmr/m/n/10 (obtained by two ap-pli-ca-tions of Eu-clid's al-go-rithm), and
 let $\OML/cmm/m/it/10 d[] \OT1/cmr/m/n/10 = [](\OML/cmm/m/it/10 d; c[]; c[]; n
\OT1/cmr/m/n/10 )$.
 []

[99]
Overfull \hbox (74.99652pt too wide) in paragraph at lines 1148--1148
[]   \OT1/cmtt/m/n/10 if(b2^2 - 4*a2*c2 != D, error("Forms must have the same d
iscriminant."));[] 
 []

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\openout2 = `chapters/ell.aux'.

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]
 6.
[103] [104] [105]
Overfull \hbox (27.74693pt too wide) in paragraph at lines 213--213
[]\OT1/cmtt/m/n/10 K<a,b,x0,x1,x2> := FieldOfFractions(PolynomialRing(Rationals
(),5));[] 
 []


Overfull \hbox (90.74638pt too wide) in paragraph at lines 213--213
[]\OT1/cmtt/m/n/10 I := ideal<R | y0^2 - (x0^3+a*x0+b), y1^2 - (x1^3+a*x1+b), y
2^2-(x2^3+a*x2+b)>;[] 
 []

[106]
Overfull \hbox (64.49661pt too wide) in paragraph at lines 213--213
[]\OT1/cmtt/m/n/10 E  := EllipticCurve([L| a,b]);  // The elliptic curve y^2 = 
x^3 + a*x + b.[] 
 []


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[109]
Overfull \hbox (27.74693pt too wide) in paragraph at lines 1213--1213
[]\OT1/cmtt/m/n/10 K<a,b,x0,x1,x2> := FieldOfFractions(PolynomialRing(Rationals
(),5));[] 
 []


Overfull \hbox (1.49716pt too wide) in paragraph at lines 1213--1213
[]\OT1/cmtt/m/n/10 // Let P0, P1, and P2 be three distinct "generic" points on 
E.[] 
 []


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[]\OT1/cmtt/m/n/10 // The algebraic formulas for the group law are built into M
AGMA.[] 
 []

[110]
Overfull \hbox (53.9967pt too wide) in paragraph at lines 1225--1225
[]\OT1/cmtt/m/n/10 ? for(x=0,4, for(y=0,4, if((y^2-(x^3+3*x+3))%5==0, print1([x
,y],"  "))))[] 
 []


Overfull \hbox (48.74675pt too wide) in paragraph at lines 1243--1243
[]\OT1/cmtt/m/n/10 ? ?ellpow    \\ type ?5 for a complete list of elliptic-curv
e functions[] 
 []


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[]\OT1/cmtt/m/n/10 ellpow(e,x,n): n times the point x on elliptic curve e (n in
 Z).[] 
 []

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 [] 
 []

[113]
Overfull \hbox (63.91136pt too wide) detected at line 1405
\OML/cmm/m/it/5 y[] \OT1/cmr/m/n/5 + \OML/cmm/m/it/5 xy \OT1/cmr/m/n/5 + \OML/c
mm/m/it/5 y \OT1/cmr/m/n/5 = \OML/cmm/m/it/5 x[] \OMS/cmsy/m/n/5 ^^@ \OT1/cmr/m
/n/5 120039822036992245303534619191166796374\OML/cmm/m/it/5 x \OT1/cmr/m/n/5 + 
504224992484910670010801799168082726759443756222911415116
 []


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[] 
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[]    \OT1/cmtt/m/n/10 http://listserv.nodak.edu/scripts/wa.exe?A2=ind0005&L=nm
brthry&P=R182[] 
 []

[114]
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\OT1/cmr/m/n/10 Robin Hood lives) has a web page, \OT1/cmtt/m/n/10 http://www.t
om.womack.net/maths/torsion.htm\OT1/cmr/m/n/10 ,
 []

[115]
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[]\OT1/cmtt/m/n/10 elltors(e,{flag=0}): torsion subgroup of elliptic curve e: o
rder, structure,[] 
 []


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[]\OT1/cmtt/m/n/10 generators. If flag = 0, use Doud's algorithm; if flag = 1, 
use Lutz-Nagell.[] 
 []

[116] [117] [118]) [119]
\openout2 = `chapters/ellapp.aux'.

 (chapters/ellapp.tex [120



]
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File: graphics/wma.eps Graphic file (type eps)
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<graphics/notes.eps>
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\OT1/cmr/m/n/10 when they aren't out thwart-ing ter-ror-ists. When Nikita in-st
alled Mi-crosoft's
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<graphics/juno.eps> [122] [123]
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[]\OT1/cmr/m/n/10 The fol-low-ing ex-am-ple is based on an ex-am-ple taken from
 Beale Screamer's
 []


Overfull \hbox (29.78421pt too wide) detected at line 136
\OML/cmm/m/it/5 rB \OT1/cmr/m/n/5 = (179671003218315746385026655733086044982194
424660\OML/cmm/m/it/5 ; \OT1/cmr/m/n/5 6978343853596863682493012826751418309351
76314718)
 []


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\OML/cmm/m/it/5 P \OT1/cmr/m/n/5 + \OML/cmm/m/it/5 r\OT1/cmr/m/n/5 (\OML/cmm/m/
it/5 nB\OT1/cmr/m/n/5 ) = (137851038548264467372645158093004000343639118915\OML
/cmm/m/it/5 ; \OT1/cmr/m/n/5 110848589228676224057229230223580815024224875699)\
OML/cmm/m/it/5 :
 []


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\OML/cmm/m/it/5 n\OT1/cmr/m/n/5 (\OML/cmm/m/it/5 rB\OT1/cmr/m/n/5 ) = \OML/cmm/
m/it/5 r\OT1/cmr/m/n/5 (\OML/cmm/m/it/5 nB\OT1/cmr/m/n/5 ) = (32890139351873263
7577115650601768681044040715701\OML/cmm/m/it/5 ; \OT1/cmr/m/n/5 586947838087815
993601350565488788846203887988162)\OML/cmm/m/it/5 :
 []


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\OML/cmm/m/it/10 P \OT1/cmr/m/n/10 = (14489646124220757767\OML/cmm/m/it/10 ;  \
OT1/cmr/m/n/10 669337780373284096274895136618194604469696830074)\OML/cmm/m/it/1
0 :
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[124]
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[]   \OT1/cmtt/m/n/10 c := ["0","1","2","3","4","5","6","7","8","9","a","b","c"
,"d","e","f"];[] 
 []


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[]\OT1/cmtt/m/n/10 E := EllipticCurve([3176890812513255034763174764138276932727
46955927,  79052896607878758718120572025718535432100651934]);[] 
 []


Overfull \hbox (258.74492pt too wide) in paragraph at lines 228--228
[]\OT1/cmtt/m/n/10 B  := E![k|771507216262649826170648268565579889907769254176,
 390157510246556628525279459266514995562533196655];[] 
 []


Overfull \hbox (253.49496pt too wide) in paragraph at lines 228--228
[]\OT1/cmtt/m/n/10 rB := E![k|179671003218315746385026655733086044982194424660,
697834385359686368249301282675141830935176314718];[] 
 []


Overfull \hbox (263.99487pt too wide) in paragraph at lines 228--228
[]\OT1/cmtt/m/n/10 PrnB := E![k|13785103854826446737264515809300400034363911891
5,110848589228676224057229230223580815024224875699];[] 
 []


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[]\OT1/cmtt/m/n/10 (14489646124220757767 : 669337780373284096274895136618194604
469696830074 : 1)[] 
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[125]
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[][] 
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[]\OT1/cmtt/m/n/10 {ispowersmooth(n, B) =   \\ true if and only if n is B-power
smooth[] 
 []

[126]
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\OT1/cmr/m/n/10 Assume that fur-ther that $\OML/cmm/m/it/10 p \OMS/cmsy/m/n/10 
^^@ \OT1/cmr/m/n/10 1$ is $\OML/cmm/m/it/10 B$\OT1/cmr/m/n/10 -power-smooth and
 let $\OML/cmm/m/it/10 m \OT1/cmr/m/n/10 = [](1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/
10 2\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 3\OML/cmm/m/it/10 ; [] B\OT1/cmr/m/n/10 
)$.
 []

[127] [128] [129]
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[]   \OT1/cmtt/m/n/10 if(isprime(p),t++;if(isalmostpowersmooth(p,B),s++,print("
BAD[] 
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[] [] 
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[]   \OT1/cmtt/m/n/10 ellpow(E,[0,1]*Mod(1,N),m);  \\ this fails if and only if
 we win![] 
 []


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[]\OT1/cmtt/m/n/10 ? N = 5959;           \\ This number motivated the ECM last 
time.[] 
 []


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[]\OT1/cmtt/m/n/10 \\ Recall what happened when we tried to factor 5959 using t
he p-1 method.[] 
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[131]
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[]\OT1/cmtt/m/n/10 [2 4]                          \\ #E(Z/101) is 16-power-smoo
th,[] 
 []


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[]\OT1/cmtt/m/n/10 [2 1]                          \\ #E(Z/59) is 29-power-smoot
h,[] 
 []


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[]\OT1/cmtt/m/n/10 E: y^2 = x^3 + 273687051132207711452727265152539544370874547
x+1,  P=[0,1][] 
 []


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[]\OT1/cmtt/m/n/10 E: y^2 = x^3 + 174264237886300715545169749498695137077020788
x+1,  P=[0,1][] 
 []


Overfull \hbox (59.24666pt too wide) in paragraph at lines 650--650
[]\OT1/cmtt/m/n/10 E: y^2 = x^3 + 652986182461202633808585244537305097270008449
x+1,  P=[0,1][] 
 []


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[]\OT1/cmtt/m/n/10 E: y^2 = x^3 + 755060727645891482095225151281965348765197238
x+1,  P=[0,1][] 
 []


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[]\OT1/cmtt/m/n/10 E: y^2 = x^3 + 722355978919416556225676818691766898771312229
x+1,  P=[0,1][] 
 []

[132]
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[]\OT1/cmtt/m/n/10 E: y^2 = x^3 + 124781379199538996805045456359983628056546634
x+1,  P=[0,1][] 
 []


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[]\OT1/cmtt/m/n/10 E: y^2 = x^3 + 350310715627251979278144271594744514052364663
x+1,  P=[0,1][] 
 []


Overfull \hbox (53.9967pt too wide) in paragraph at lines 650--650
[]\OT1/cmtt/m/n/10 E: y^2 = x^3 + 39638500146503230913823829562620410547947307x
+1,  P=[0,1][] 
 []


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[]                            \OT1/cmtt/m/n/10 80061047060165522139279418005808
8102053408423).[] 
 []


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[]  \OT1/cmtt/m/n/10 ***   impossible inverse modulo: Mod(1406051123, 100432032
2301182911).[] 
 []


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[]                                        \OT1/cmtt/m/n/10 79716645459013454877
3760793).[] 
 []


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\OML/cmm/m/it/10 N \OT1/cmr/m/n/10 = \OML/cmm/m/it/10 N[] \OMS/cmsy/m/n/10 ^^A 
\OML/cmm/m/it/10 N[] \OMS/cmsy/m/n/10 ^^A \OML/cmm/m/it/10 N[] \OMS/cmsy/m/n/10
 ^^A \OML/cmm/m/it/10 N[] \OT1/cmr/m/n/10 = 1406051123 \OMS/cmsy/m/n/10 ^^A \OT
1/cmr/m/n/10 714284357 \OMS/cmsy/m/n/10 ^^A \OT1/cmr/m/n/10 2029256729 \OMS/cms
y/m/n/10 ^^A \OT1/cmr/m/n/10 392836669307471617\OML/cmm/m/it/10 ;
 []


Overfull \hbox (22.49698pt too wide) in paragraph at lines 687--687
[]                       \OT1/cmtt/m/n/10 Mod(63845045623767003, 39283666930747
1617)][] 
 []


Overfull \hbox (43.4968pt too wide) in paragraph at lines 687--687
[]  \OT1/cmtt/m/n/10 ***   impossible inverse modulo: Mod(615433499, 3928366693
07471617).[] 
 []

[133] [134] [135] [136]
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File: graphics/wiles.eps Graphic file (type eps)
<graphics/wiles.eps> [137]) [138]
\openout2 = `chapters/bsd.aux'.

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]
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[]\OT1/cmtt/m/n/10 http://www.msri.org/publications/ln/hosted/cmi/2000/cmiparis
/index-tate.html[] 
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[]  $[]$ 
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\OT1/cmr/m/it/10 NOT \OT1/cmr/m/n/10 nu-mer-i-cal in-te-gra-tion. In PARI, $
[]$ is ap-prox-i-mated by \OT1/cmtt/m/n/10 E.omega[1]*2^(E.disc>0)\OT1/cmr/m/n/
10 . 
 []

[144]
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[]\OT1/cmtt/m/n/10 ? L = (1+eps) * sum(n=1,100, ellak(E,n)/n * exp(-2*Pi*n/sqrt
(N)))[] 
 []

[145]
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[] 
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[146]
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[]\OT1/cmtt/m/n/10 ? r(E,s) = L1=elllseries(E,s); L2=elllseries(E,s^2-s+1); (L2
-L1)/((s-1)*L1);[] 
 []


Overfull \hbox (0.30826pt too wide) in paragraph at lines 544--555
\OT1/cmr/m/n/10 It cer-tainly looks like $[][] \OML/cmm/m/it/10 r\OT1/cmr/m/n/1
0 (\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 ) = 4$. We know for a fact that $[][] \OML
/cmm/m/it/10 r\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 ) \OMS/cmsy/m/
n/10 2
 []


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[]\OT1/cmtt/m/n/10 ? r(E,s) = L1=elllseries(E,s); L2=elllseries(E,s^2-s+1); (L2
-L1)/((s-1)*L1);[] 
 []

) [147]
\openout2 = `chapters/pari.aux'.

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]
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[149]
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 []\OT1/cmtt/m/n/10 ? gb(n) = forprime(p=2,n,if(isprime(n-p),return([p,n-p])));
[] 
 []

[150]
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[]  \OT1/cmtt/m/n/10 http://www.faber.co.uk/faber/million_dollar.asp?PGE=&ORD=f
aber&TAG=&CID=[] 
 []


Overfull \hbox (48.74675pt too wide) in paragraph at lines 125--125
[]  \OT1/cmtt/m/n/10 http://www.informatik.uni-giessen.de/staff/richstein/ca/Go
ldbach.html[] 
 []


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[]  \OT1/cmtt/m/n/10 http://perso.wanadoo.fr/yves.gallot/primes/chrrcds.html#tw
in[] 
 []

[152]
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[]                \OT1/cmtt/m/n/10 i686 running linux (ix86 kernel) 32-bit vers
ion[] 
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[]                \OT1/cmtt/m/n/10 (readline v4.2 enabled, extended help availa
ble)[] 
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[]\OT1/cmtt/m/n/10 PARI/GP is free software, covered by the GNU General Public 
License, and[] 
 []

[153]
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[]\OT1/cmtt/m/n/10 Type ?12 for how to get moral (and possibly technical) suppo
rt.[] 
 []


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[]\OT1/cmtt/m/n/10 %6 = 1378425503766585493035778406754125077322291549582802091
3935[] 
 []


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[]\OT1/cmtt/m/n/10 845011397194393261309756046226816251290119446623115998366224
1797[] 
 []


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[]  \OT1/cmtt/m/n/10 0: list of user-defined identifiers (variable, alias, func
tion)[] 
 []

[154]
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[]  \OT1/cmtt/m/n/10 ??             (opens the full user's manual in a dvi prev
iewer)[] 
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[]  \OT1/cmtt/m/n/10 ??  keyword    (long help text about "keyword" from the us
er's manual)[] 
 []


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[]\OT1/cmtt/m/n/10 addprimes     bestappr      bezout        bezoutres     bigo
mega[] 
 []


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[]\OT1/cmtt/m/n/10 binomial      chinese       content       contfrac      cont
fracpnqn[] 
 []


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[]\OT1/cmtt/m/n/10 core          coredisc      dirdiv        direuler      dirm
ul[] 
 []


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[]\OT1/cmtt/m/n/10 divisors      eulerphi      factor        factorback    fact
orcantor[] 
 []


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[]\OT1/cmtt/m/n/10 factorff      factorial     factorint     factormod     ffin
it[] 
 []


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[]\OT1/cmtt/m/n/10 fibonacci     gcd           hilbert       isfundamental ispr
ime[] 
 []


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[]\OT1/cmtt/m/n/10 moebius       nextprime     numdiv        omega         prec
prime[] 
 []


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[]\OT1/cmtt/m/n/10 prime         primes        qfbclassno    qfbcompraw    qfbh
classno[] 
 []


Overfull \hbox (1.49716pt too wide) in paragraph at lines 276--276
[]\OT1/cmtt/m/n/10 qfbnucomp     qfbnupow      qfbpowraw     qfbprimeform  qfbr
ed[] 
 []


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[]\OT1/cmtt/m/n/10 quadclassunit quaddisc      quadgen       quadhilbert   quad
poly[] 
 []


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[]\OT1/cmtt/m/n/10 sqrtint       znlog         znorder       znprimroot    znst
ar[] 
 []


Overfull \hbox (74.99652pt too wide) in paragraph at lines 276--276
[]\OT1/cmtt/m/n/10 gcd(x,y,{flag=0}): greatest common divisor of x and y. flag 
is optional, and[] 
 []


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[]\OT1/cmtt/m/n/10 can be 0: default, 1: use the modular gcd algorithm (x and y
 must be[] 
 []


Overfull \hbox (80.24648pt too wide) in paragraph at lines 276--276
[]\OT1/cmtt/m/n/10 polynomials), 2 use the subresultant algorithm (x and y must
 be polynomials).[] 
 []


Overfull \hbox (90.74638pt too wide) in paragraph at lines 276--276
[]\OT1/cmtt/m/n/10 \\ if set up correctly, brings up the typeset subsection fro
m the manual on gcd[] 
 []


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\OT1/cmr/m/n/10 the last file loaded). For ex-am-ple, sup-pose we change \OT1/c
mtt/m/n/10 return (lift(Mod(a,p)^n))
 []

[156]
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[]\OT1/cmtt/m/n/10 %13 = 6                    \\ ouch!! what have you done to m
y eye![] 
 []

[157]
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[]\OT1/cmtt/m/n/10 input(): read an expression from the input file or standard 
input.[] 
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[]\OT1/cmtt/m/n/10 ? s                  \\ s is not the string "1+1", as you mi
ght expect[] 
 []


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[]\OT1/cmtt/m/n/10 ? type(s)            \\ PARI views s as a polynomial in the 
variable hithere[] 
 []

[158]
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[]\OT1/cmtt/m/n/10 qfbclassno(x,{flag=0}): class number of discriminant x using
 Shanks's[] 
 []


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[]\OT1/cmtt/m/n/10 method by default. If (optional) flag is set to 1, use Euler
 products.[] 
 []


Overfull \hbox (48.74675pt too wide) in paragraph at lines 474--474
[]\OT1/cmtt/m/n/10 ? qfbclassno(-15,1)  \\ ALWAYS use flag=1, since ``the autho
rs were too[] 
 []


Overfull \hbox (53.9967pt too wide) in paragraph at lines 474--474
[]\OT1/cmtt/m/n/10 %32 = 2              \\ lazy to implement Shanks' method com
pletely...''[] 
 []


Overfull \hbox (59.24666pt too wide) in paragraph at lines 474--474
[]\OT1/cmtt/m/n/10 ? a=-11/9;b=28/27;          \\ this is an ``amazing'' point 
on the curve.[] 
 []

[159]
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[]\OT1/cmtt/m/n/10 ? elladd(E,P,[1,0])    \\ nonsense, since [1,0] isn't even o
n E!!![] 
 []

[160]
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[]\OT1/cmtt/m/n/5 4660645813671121765025590267647300672252945873586541077711389
394563791/920992883734992462745141522111225908861976098219465616585649245395649
][] 
 []


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[]\OT1/cmtt/m/n/10 ? for(x=0,4, for(y=0,4, if(ellisoncurve(E5,[x,y]),print([x,y
]))))[] 
 []


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[]\OT1/cmtt/m/n/10 ? 5+1 - 9          \\ 8 points above, plus the point at infi
nity[] 
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[161]
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[162]
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\OT1/cmr/m/n/10 MAGMA also of-fers so-phis-ti-cated fea-tures for com-put-ing w
ith el-lip-tic curves,
 []

) [165]
\openout2 = `chapters/solutions.aux'.

 (chapters/solutions.tex [166



]
 10.
[167] [168]
Overfull \hbox (10.30612pt too wide) in paragraph at lines 178--183
[]\OT1/cmr/m/n/10 Using a com-puter, we find that $5352381469067 = 141307 \OMS/
cmsy/m/n/10 ^^A \OT1/cmr/m/n/10 37877681$,
 []

[169] [170] [171] [172]
Overfull \hbox (7.17534pt too wide) in paragraph at lines 422--428
[]\OT1/cmr/m/n/10 First we com-pute $\OML/cmm/m/it/10 ^^K \OT1/cmr/m/n/10 = [[]
]$. This gives $\OML/cmm/m/it/10 ^^K \OT1/cmr/m/n/10 = 1 + [][]$.
 []


Overfull \hbox (18.38747pt too wide) in paragraph at lines 429--435
[]\OT1/cmr/m/n/10 This is $[[]][]$. As above, if $\OML/cmm/m/it/10 ^^K \OT1/cmr
/m/n/10 = [[]]$ then $\OML/cmm/m/it/10 ^^K \OT1/cmr/m/n/10 = 1 + [][]$.
 []

[173]
Overfull \hbox (21.24724pt too wide) in paragraph at lines 489--489
 []  \OT1/cmtt/m/n/10 [7,  2,1,1,3,18,   5,1,1,6,30,   8,1,1,9,42,  11,1,1,12,5
4,[] 
 []

[174] [175] [176] [177]
Overfull \hbox (1.97264pt too wide) in paragraph at lines 721--728
\OT1/cmr/m/n/10 [1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1/c
mr/m/n/10 2504]$, and from this we find that $[10\OML/cmm/m/it/10 ; \OMS/cmsy/m
/n/10 ^^@\OT1/cmr/m/n/10 5\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 251]$ and $[10\OML
/cmm/m/it/10 ; \OT1/cmr/m/n/10 5\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 251]$
 []

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[181] (main.aux (chapters/preface.aux) (chapters/introduction.aux)
(chapters/primes.aux) (chapters/crypto.aux) (chapters/reciprocity.aux)
(chapters/contfrac.aux) (chapters/quadform.aux) (chapters/ell.aux)
(chapters/ellapp.aux) (chapters/bsd.aux) (chapters/pari.aux)
(chapters/solutions.aux))

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