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Greetings, every mathematicians! I'm a foreigner (meaning English is not my first language) and an undergraduate student. I'm currently studying linear algebra, set theory and have already studied number theory, and got a very good grade on the class!

However, the textbook we used on that number theory class was very poor (what a shame...). I used to love number theory so the only obstacle for that class was that poor textbook. Now, even though I got good grade on that class, my brain is messed up with all the number theory stuff - it is not well organized, so I want you to recommend me a suitable textbook.

The conditions are (1) I have to review the stuff so that all the theorems should be well organized in my brain (2) Since I've already took the class, I want a little bit more challenge. The problem is that I haven't took that many classes such as analysis or other algebra. So I doubt if there is a suitable book that can satisfy both conditions.

I have already checked possible recommendations here. But there were just so many recommendations so I couldn't find which suits me best.

I hope this question doesn't violate the rules in this forum! Thanks in advance.

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  • 1
    $\begingroup$ G.H.Hardy and Wright's textbook is always a good one. $\endgroup$ – A. Wong Apr 5 '14 at 0:28
  • 1
    $\begingroup$ It will be helpful to explain what level / difficulty of number theory you are learning, and what your current textbook is. $\endgroup$ – Calvin Lin Apr 5 '14 at 0:58
  • $\begingroup$ I wanted to, but the textbook I used was written only in my first language, so you probably haven't even heard about it :( It deals with the general solution for $f(x) \equiv 0 \pmod {p^{a}}$, primitive roots, index, Legendre symbol, quadratic reciprocity law, Mobius inversion formula, number that can be expressed as the sum of n square(third power and etc.) numbers, Pythagorean number, Fermat's last theorem(of course, for some special cases...), fibonacci number, continued fraction, pell's equation. $\endgroup$ – Taxxi Apr 5 '14 at 1:12
  • $\begingroup$ I'm so sorry for the inconvenience, but I think this is the only way to describe what I've learned from the textbook. It didn't deal with the general 'ring' or something, and I haven't learned about that concept (only field from linear algebra...). But I'm quite confident that if the textbook is written well, then I can study by myself. Again, sorry for the inconvenience, and I'll try to answer any question that might arise for all of you. $\endgroup$ – Taxxi Apr 5 '14 at 1:17
  • $\begingroup$ @A.Wong Thanks. I'm considering buying the book for reviewing. I was a little bit surprised the book has quite a bit of history (published in the early days). $\endgroup$ – Taxxi Apr 5 '14 at 4:11
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Take a look at how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $ (x,y)$ . It seems to me you are in good position to learn all the ingredients. I can also tell you that many students who take algebraic number theory have no idea how to solve such problems. So, my general feeling is that a course, maybe half-length, in integral binary quadratic forms, is a good lead-in to algebraic number theory.

And, since you know fields and matrices but not rings, it is probably a good choice at this point. Two books are involved, Buell and Conway. I have put quite a number of answers on this site in this style; sometimes the Conway topograph only, sometimes the Lagrange cycle method only...

Sample:

Pell x^2 -  12553 y^2.

0  form   1 224 -9   delta  -24
1  form   -9 208 193   delta  1
2  form   193 178 -24   delta  -8
3  form   -24 206 81   delta  2
4  form   81 118 -112   delta  -1
5  form   -112 106 87   delta  1
6  form   87 68 -131   delta  -1
7  form   -131 194 24   delta  8
8  form   24 190 -147   delta  -1
9  form   -147 104 67   delta  2
10  form   67 164 -87   delta  -2
11  form   -87 184 47   delta  4
12  form   47 192 -71   delta  -2
13  form   -71 92 147   delta  1
14  form   147 202 -16   delta  -13
15  form   -16 214 69   delta  3
16  form   69 200 -37   delta  -5
17  form   -37 170 144   delta  1
18  form   144 118 -63   delta  -2
19  form   -63 134 128   delta  1
20  form   128 122 -69   delta  -2
21  form   -69 154 96   delta  1
22  form   96 38 -127   delta  -1
23  form   -127 216 7   delta  31
24  form   7 218 -96   delta  -2
25  form   -96 166 59   delta  3
26  form   59 188 -63   delta  -3
27  form   -63 190 56   delta  3
28  form   56 146 -129   delta  -1
29  form   -129 112 73   delta  2
30  form   73 180 -61   delta  -3
31  form   -61 186 64   delta  3
32  form   64 198 -43   delta  -4
33  form   -43 146 168   delta  1
34  form   168 190 -21   delta  -9
35  form   -21 188 177   delta  1
36  form   177 166 -32   delta  -6
37  form   -32 218 21   delta  10
38  form   21 202 -112   delta  -1
39  form   -112 22 111   delta  1
40  form   111 200 -23   delta  -9
41  form   -23 214 48   delta  4
42  form   48 170 -111   delta  -1
43  form   -111 52 107   delta  1
44  form   107 162 -56   delta  -3
45  form   -56 174 89   delta  2
46  form   89 182 -48   delta  -4
47  form   -48 202 49   delta  4
48  form   49 190 -72   delta  -2
49  form   -72 98 141   delta  1
50  form   141 184 -29   delta  -7
51  form   -29 222 8   delta  27
52  form   8 210 -191   delta  -1
53  form   -191 172 27   delta  7
54  form   27 206 -72   delta  -2
55  form   -72 82 151   delta  1
56  form   151 220 -3   delta  -74
57  form   -3 224 3   delta  74
58  form   3 220 -151   delta  -1
59  form   -151 82 72   delta  2
60  form   72 206 -27   delta  -7
61  form   -27 172 191   delta  1
62  form   191 210 -8   delta  -27
63  form   -8 222 29   delta  7
64  form   29 184 -141   delta  -1
65  form   -141 98 72   delta  2
66  form   72 190 -49   delta  -4
67  form   -49 202 48   delta  4
68  form   48 182 -89   delta  -2
69  form   -89 174 56   delta  3
70  form   56 162 -107   delta  -1
71  form   -107 52 111   delta  1
72  form   111 170 -48   delta  -4
73  form   -48 214 23   delta  9
74  form   23 200 -111   delta  -1
75  form   -111 22 112   delta  1
76  form   112 202 -21   delta  -10
77  form   -21 218 32   delta  6
78  form   32 166 -177   delta  -1
79  form   -177 188 21   delta  9
80  form   21 190 -168   delta  -1
81  form   -168 146 43   delta  4
82  form   43 198 -64   delta  -3
83  form   -64 186 61   delta  3
84  form   61 180 -73   delta  -2
85  form   -73 112 129   delta  1
86  form   129 146 -56   delta  -3
87  form   -56 190 63   delta  3
88  form   63 188 -59   delta  -3
89  form   -59 166 96   delta  2
90  form   96 218 -7   delta  -31
91  form   -7 216 127   delta  1
92  form   127 38 -96   delta  -1
93  form   -96 154 69   delta  2
94  form   69 122 -128   delta  -1
95  form   -128 134 63   delta  2
96  form   63 118 -144   delta  -1
97  form   -144 170 37   delta  5
98  form   37 200 -69   delta  -3
99  form   -69 214 16   delta  13
100  form   16 202 -147   delta  -1
101  form   -147 92 71   delta  2
102  form   71 192 -47   delta  -4
103  form   -47 184 87   delta  2
104  form   87 164 -67   delta  -2
105  form   -67 104 147   delta  1
106  form   147 190 -24   delta  -8
107  form   -24 194 131   delta  1
108  form   131 68 -87   delta  -1
109  form   -87 106 112   delta  1
110  form   112 118 -81   delta  -2
111  form   -81 206 24   delta  8
112  form   24 178 -193   delta  -1
113  form   -193 208 9   delta  24
114  form   9 224 -1   delta  -224
115  form   -1 224 9   delta  24
116  form   9 208 -193   delta  -1
117  form   -193 178 24   delta  8
118  form   24 206 -81   delta  -2
119  form   -81 118 112   delta  1
120  form   112 106 -87   delta  -1
121  form   -87 68 131   delta  1
122  form   131 194 -24   delta  -8
123  form   -24 190 147   delta  1
124  form   147 104 -67   delta  -2
125  form   -67 164 87   delta  2
126  form   87 184 -47   delta  -4
127  form   -47 192 71   delta  2
128  form   71 92 -147   delta  -1
129  form   -147 202 16   delta  13
130  form   16 214 -69   delta  -3
131  form   -69 200 37   delta  5
132  form   37 170 -144   delta  -1
133  form   -144 118 63   delta  2
134  form   63 134 -128   delta  -1
135  form   -128 122 69   delta  2
136  form   69 154 -96   delta  -1
137  form   -96 38 127   delta  1
138  form   127 216 -7   delta  -31
139  form   -7 218 96   delta  2
140  form   96 166 -59   delta  -3
141  form   -59 188 63   delta  3
142  form   63 190 -56   delta  -3
143  form   -56 146 129   delta  1
144  form   129 112 -73   delta  -2
145  form   -73 180 61   delta  3
146  form   61 186 -64   delta  -3
147  form   -64 198 43   delta  4
148  form   43 146 -168   delta  -1
149  form   -168 190 21   delta  9
150  form   21 188 -177   delta  -1
151  form   -177 166 32   delta  6
152  form   32 218 -21   delta  -10
153  form   -21 202 112   delta  1
154  form   112 22 -111   delta  -1
155  form   -111 200 23   delta  9
156  form   23 214 -48   delta  -4
157  form   -48 170 111   delta  1
158  form   111 52 -107   delta  -1
159  form   -107 162 56   delta  3
160  form   56 174 -89   delta  -2
161  form   -89 182 48   delta  4
162  form   48 202 -49   delta  -4
163  form   -49 190 72   delta  2
164  form   72 98 -141   delta  -1
165  form   -141 184 29   delta  7
166  form   29 222 -8   delta  -27
167  form   -8 210 191   delta  1
168  form   191 172 -27   delta  -7
169  form   -27 206 72   delta  2
170  form   72 82 -151   delta  -1
171  form   -151 220 3   delta  74
172  form   3 224 -3   delta  -74
173  form   -3 220 151   delta  1
174  form   151 82 -72   delta  -2
175  form   -72 206 27   delta  7
176  form   27 172 -191   delta  -1
177  form   -191 210 8   delta  27
178  form   8 222 -29   delta  -7
179  form   -29 184 141   delta  1
180  form   141 98 -72   delta  -2
181  form   -72 190 49   delta  4
182  form   49 202 -48   delta  -4
183  form   -48 182 89   delta  2
184  form   89 174 -56   delta  -3
185  form   -56 162 107   delta  1
186  form   107 52 -111   delta  -1
187  form   -111 170 48   delta  4
188  form   48 214 -23   delta  -9
189  form   -23 200 111   delta  1
190  form   111 22 -112   delta  -1
191  form   -112 202 21   delta  10
192  form   21 218 -32   delta  -6
193  form   -32 166 177   delta  1
194  form   177 188 -21   delta  -9
195  form   -21 190 168   delta  1
196  form   168 146 -43   delta  -4
197  form   -43 198 64   delta  3
198  form   64 186 -61   delta  -3
199  form   -61 180 73   delta  2
200  form   73 112 -129   delta  -1
201  form   -129 146 56   delta  3
202  form   56 190 -63   delta  -3
203  form   -63 188 59   delta  3
204  form   59 166 -96   delta  -2
205  form   -96 218 7   delta  31
206  form   7 216 -127   delta  -1
207  form   -127 38 96   delta  1
208  form   96 154 -69   delta  -2
209  form   -69 122 128   delta  1
210  form   128 134 -63   delta  -2
211  form   -63 118 144   delta  1
212  form   144 170 -37   delta  -5
213  form   -37 200 69   delta  3
214  form   69 214 -16   delta  -13
215  form   -16 202 147   delta  1
216  form   147 92 -71   delta  -2
217  form   -71 192 47   delta  4
218  form   47 184 -87   delta  -2
219  form   -87 164 67   delta  2
220  form   67 104 -147   delta  -1
221  form   -147 190 24   delta  8
222  form   24 194 -131   delta  -1
223  form   -131 68 87   delta  1
224  form   87 106 -112   delta  -1
225  form   -112 118 81   delta  2
226  form   81 206 -24   delta  -8
227  form   -24 178 193   delta  1
228  form   193 208 -9   delta  -24
229  form   -9 224 1   delta  224
230  form   1 224 -9

 disc   50212
Automorph, written on right of Gram matrix:  
6569654811042976036551664287776695086862221675257028799978001629441955413114961078437403743404915644466809863876018081  1471866589689666239349905729201246719313821836636303758803979047726858179020247176803043556748067158933333371566607194904
163540732187740693261100636577916302145980204070700417644886560858539797668916352978115950749785239881481485729623021656  36639693664864958266523094257741028375786427933512150581254567633942356633250378028176410371695298649096319613299432869025


 Pell automorph 
18323131659838000621279822961014402535436645077593703805027272817785899294331746494627423887719351782370393211581654443553  2052926811152708922506596290962583340838489501699502342696260998457250080137906978934289529762054116232237090363957790847768
163540732187740693261100636577916302145980204070700417644886560858539797668916352978115950749785239881481485729623021656  18323131659838000621279822961014402535436645077593703805027272817785899294331746494627423887719351782370393211581654443553

Pell unit 
18323131659838000621279822961014402535436645077593703805027272817785899294331746494627423887719351782370393211581654443553^2 - 12553 * 163540732187740693261100636577916302145980204070700417644886560858539797668916352978115950749785239881481485729623021656^2 = 1 

=========================================

Pell NEGATIVE 
3026807861414232601576476947093931340883926646932707209616924^2 - 12553 * 27015380505739903416049453225901131788373049465835213058397^2 = -1 

=========================================

12553       12553

=========================

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  • $\begingroup$ Thanks for your kind answer. I've looked into that question and I realized I haven't learned the way to add 'root' into the quadratic residue (Dirichlet's formula, is it right?). My limit of understading can be seen here, and I understand the answer given there though, but that's not as general as the question you've linked. How to solve that question seems so much interesting! I'm eager to learn that kind of method. $\endgroup$ – Taxxi Apr 5 '14 at 2:23
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    $\begingroup$ The two I use in the other answer, Binary Quadratic Forms by Duncan A. Buell, and The Sensual Quadratic Form by John H. Conway. You can see dozens of examples I worked out on this site. $\endgroup$ – Will Jagy Apr 5 '14 at 2:28
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    $\begingroup$ @TaxxiDriver Be aware that the books recommended by Will are devoted mostly to number theory of binary quadratic forms and related theory. If you are interested in general textbooks on elementary number theory then you will need to look elsewhere. $\endgroup$ – Bill Dubuque Apr 5 '14 at 18:14
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    $\begingroup$ @TaxxiDriver Those and more can be downloaded from the site linked here. $\endgroup$ – Bill Dubuque Apr 5 '14 at 20:41
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    $\begingroup$ @TaxxiDriver, well, the Conway book is reasonable, maa.org/publications/ebooks/the-sensual-quadratic-form The other main item you rarely see, the Lagrange method as an alternative to continued fractions especially suited for number theory, is also in Introduction to the Theory of Numbers by Leonard Eugene Dickson (1929), three chapters out of 11, see especially chapter 7, pages 99-116. This was re-issued in paperback, you will only be able to find it used, but inexpensive. abebooks.com/servlet/… $\endgroup$ – Will Jagy Apr 5 '14 at 21:08
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It wasn't clear to me whether you wanted a book that covered what you've already learnt (but in a clearer manner) or whether you want something to build on what you've learnt.

If it is the latter you want, then I recommend A classical introduction to modern number theory, by Ireland and Rosen. It is a beautiful book, that begins with equations mod $p$, Gauss sums, and related (fairly elementary) topics, and builds up to a treatment of the basic theorems from algebraic number theory (but in a manner which requires essentially no prior algebra background, if my memory is correct).

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  • $\begingroup$ I wish one of my friend owned a bookstore... I'm aiming at both the two objects you've mentioned, and I'm realizing it's hard to achieve the both just using one book. I've also seen a lot of recommendations about the book. I'll definitely have a look at it. Thanks. $\endgroup$ – Taxxi Apr 6 '14 at 1:29

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