I need an intermediate number theory book

[lewelma]

My ds has finished the AoPS intro number theory book and the diophantine equations chapter in Volume 2: and beyond.  AoPS does not have an intermediate book, so does anyone have any suggestions for intermediate number theory?  Although this is specifically for prepping for the NZMO and IMO, I still want it to be a systematic approach to improving his skills and knowledge.  Meaning that I really don't just want a bunch of previous math olympiad problems, but I also want him to study material that will be useful for the exams.  I'm open to suggestions!

 

Thanks,

 

Ruth in NZ

[onaclairadeluna]

My son really enjoyed the EPGY number theory class. They used the Stark book but they also have lectures and an interactive computer thing (really old) that walks you through the material. It was good. I don't know how this would correlate with competition math.

 

http://epgy.stanford.edu/courses/math/M152/

 

[Kathy in Richmond]

Yes to what onaclairadeluna said. My daughter also enjoyed the EPGY course and text.

 

A couple of classic number theory texts for his shelf: (very expensive; maybe you can locate a used copy)

 

Niven, Zuckerman, & Montgomery, An Introduction to the Theory of Numbers

Hardy & Wright, An Introduction to the Theory of Numbers

 

Nice to see you back, Ruth.  Hope that the Olympiad study is going well for your ds!

[Matryoshka]

My ds has finished the AoPS intro number theory book and the diophantine equations chapter in Volume 2: and beyond.  AoPS does not have an intermediate book, so does anyone have any suggestions for intermediate number theory?  Although this is specifically for prepping for the NZMO and IMO, I still want it to be a systematic approach to improving his skills and knowledge.  Meaning that I really don't just want a bunch of previous math olympiad problems, but I also want him to study material that will be useful for the exams.  I'm open to suggestions!

AoPS doesn't have a book for intermediate number theory, but they do have an online course...  if you're about 12ish hours off from EST in NZ(??) the timing might not even be too bad...

[lewelma]

Thanks everyone!  I will check out the EPGY and AoPS classes and see what I think.  He has never taken an online class, so it would be interesting to see what he thinks of the idea.

 

Kathy, I think that the study has gone well.  He has finished the counting and number theory books.  And has gotten through the Diophantine equations section last week, and has also practiced counting problems every week after finishing the book in April.  I had him skip ahead in the geometry book to get in the analytical geometry section which seems to be a favorite on the exam.  He has been reviewing the intro algebra book which he finished a year ago, but has not finished his review which I am a bit concerned about -- he did some of that material at a very young age, and I don't know how much really stuck.  He has also gotten through the first 5 chapters of the Zeitz book, and feels confident about proof writing.  Finally, he has practiced his proof writing on previous exam problems and they look pretty good to me (but I'm pretty sure that I am not a good judge).  I think that the exam is about to be posted, because the IMO finished yesterday and the team and coaches are coming back today.  I have been in contact with the leader by e-mail and DS has interviewed with the deputy leader here in town.  So really, I think he is either ready, or not, because this is just about all that I can do.  If he doesn't make it this year, he has many more years to try again. He understands that the competition is fierce although the choice for camp attendance is subjective and his young age will definitely help.

 

Thanks for all your help!

 

Ruth in NZ

[mathwonk]

here are cheaper used copies of those books Kathy listed (older editions, just as good).

 

http://www.abebooks.com/servlet/SearchResults?an=niven%2C+zuckerman&sts=t

 

 

http://www.abebooks.com/servlet/SearchResults?an=hardy%2C+wright&sts=t&tn=number+theory

 

 

and another good elementary book that might still be useful:

 

http://www.abebooks.com/servlet/SearchResults?an=vanden+eynden&tn=number+theory

 

 

And here is a historical approach that looks interesting by the rgeat great Andre Weil:

 

http://www.abebooks.com/servlet/SearchResults?an=weil&sts=t&tn=number+theory

 

 

 

You do NOT want Weil's "Basic number theory" which uses Fourier analysis and Haar measure on compact groups as a background tool, hence extremely advanced. But his "Number theory for beginners" might interest.

 

Hereb is the page of recommended alternative books for a course on number theory at Berkeley from a famous expert, Ken Ribet:

 

http://math.berkeley.edu/~ribet/Math115/other_books.html

 

 

Seeing those I am tempted to recommend Gauss' Disquisitiones Arithmeticae, but it may be pricey.

 

http://www.abebooks.com/servlet/SearchResults?an=gauss&sts=t&tn=disquisitiones+arithmeticae

 

http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Dstripbooks&field-keywords=gauss%2C+disquisitiones+arithmeticae

 

Be careful, some of these are in Latin and some in German.

 

 

The first four sections of the free math 4000 notes on my webpage are in a sense an intermediate course in number theory. They include the proof of Fermat's that every prime number of form 4K+1 is a sum of two squares, e.g. 5 = 1^2 _+ 2^2, 17 = 4^1 + 1^1, 29 = 2^2 + 5^2, 41 = 5^2 + 4^2, 61 = 5^2 + 6^2,101 = 10^2 + 1^2, 173 = 2^2 + (13)^2 ...

[Kathy in Richmond]

Ruth, if your son made it through all that you listed, then I'd guarantee that his basic algebra is strong. He sounds incredibly well-prepared for the olympiad exam at this point. Best of luck to him! I'll be cheering for him from Richmond & can't wait to hear how it goes!

[lewelma]

Thanks Mathwonk for all the links, I will go look them up!

 

Kathy, thanks for your encouragement.  I am mostly concerned about the more advanced topics in the back of the algebra book like infinite series and quadratic optimization.  I think he could definitely see series on the exam, or need to use them in some way for his proofs.  But he could always look them up, I guess.

 

My next big project is trying to design a program of study for the up coming year that incorporates review, proof writing, and math olympiad questions into his standard study.  What I really need is a math olympiad club here in town, but there just isn't one.  He is taking math as an elective next year, so he will have 2.5 hours per day to put towards it.  So kind of like 10 1.25 hour periods per week.

 

My current thought is to have him do

3 periods a week of Intermediate Algebra,

3 periods a week of Geometry.

2 periods each week for reviewing/working MO questions for number theory and counting.

2 periods each week for Zeitz, both the book and the lecture series. 

 

So basically I need to find and organize the material for him to work on for 4 periods a week.  So what I really need is to take a couple of weekends and sort through all my links and print what I like that appears to be at his level.  I hope to do this while he is working on the the exam this month. 

 

[mathwonk]

Your child might benefit from this book by my colleague ted Shifrin:

 

http://www.abebooks.com/servlet/SearchResults?an=shifrin&sts=t&tn=abstract+algebra

 

He is a highly decorated teacher and math contest coach.

 

 

My free math 4000 notes are based on this book, but not as well written, and lack his wonderful problems.

[Kathy in Richmond]

I am mostly concerned about the more advanced topics in the back of the algebra book like infinite series and quadratic optimization.  I think he could definitely see series on the exam, or need to use them in some way for his proofs.  But he could always look them up, I guess.

 

Ok, I see what you mean now. Infinite series often make an appearance in some guise or another on olympiad problems in the US, and it's handy to know them well. But if the junior olympiad test in NZ is take-home like the USAMTS, he should be fine looking up relevant stuff as he works.

 

My next big project is trying to design a program of study for the up coming year that incorporates review, proof writing, and math olympiad questions into his standard study.  What I really need is a math olympiad club here in town, but there just isn't one.  He is taking math as an elective next year, so he will have 2.5 hours per day to put towards it.  So kind of like 10 1.25 hour periods per week.

 

My current thought is to have him do

3 periods a week of Intermediate Algebra,

3 periods a week of Geometry.

2 periods each week for reviewing/working MO questions for number theory and counting.

2 periods each week for Zeitz, both the book and the lecture series. 

 

So basically I need to find and organize the material for him to work on for 4 periods a week.  So what I really need is to take a couple of weekends and sort through all my links and print what I like that appears to be at his level.  I hope to do this while he is working on the the exam this month.

 

Looks like fun!

 

If he does make the training camp this time around, the coaches may give your son problems and materials to work through during the academic year, too.

 

When my kids were studying for olympiad & math team, we had a similar time schedule. Everyone's different in how they like to approach it. Around here, it was useful to spend about half of that time working on assorted problems from past AMC exams and other contests like USAMTS, and to use the other half for a devoted study of one math area at a time. We'd have a season (8 to 12 weeks) of number theory, then another of geometry, etc. They liked to get really involved & in depth in one area at a time. But again, YMMV!

 

Good luck. I miss those days! :001_smile:

[lewelma]

Kathy, I really like your approach to keeping it all fresh in their heads while still allowing them to focus on one thing at a time. 8 to 12 weeks per topic sounds really quite interesting. My initial approach was standard American: a year of algebra, a year of geometry, etc; but I quickly realized that he was going to forget stuff, so I started thinking about doing 2 topics at once.  However, I know he would much prefer to focus on just one topic at a time, but I just never thought of doing a term of this and then a term of that.  Excellent idea. NZ has 4 terms a year, do you think a term of algebra, geo, num theo, counting would work?  It would take him forever to get through that intermediate algebra book.

 

I've been looking at all the number theory books that you and mathwonk have suggested, and I will have to say that the upper undergrad rating on most of them is scaring me off.  Is that really where my child is at?!?!?! Or are these books set up like Zeitz? Where you can work up to your level and then move to the next chapter, studying the book in many passes. The AoPS intermediate course looks good, and the timing is pretty good too.  So I might ask him if he would like to go that route.  He can't do the EPGY classes apparently without taking some sort of test, none of which look to be easily accessible to us.  sigh.

 

As for the camp, apparently of the 26 kids going to the camp, they choose 12 to make up the squad.  The squad is then given intensive training.  I think ds is too young for the squad, but the deputy leader did say he could mentor ds starting in September.  I just am not sure what 'mentor' means exactly.  Handing ds books and problems to do would be awesome.  So here's to hoping!

 

 

[Kathy in Richmond]

Kathy, I really like your approach to keeping it all fresh in their heads while still allowing them to focus on one thing at a time. 8 to 12 weeks per topic sounds really quite interesting. My initial approach was standard American: a year of algebra, a year of geometry, etc; but I quickly realized that he was going to forget stuff, so I started thinking about doing 2 topics at once.  However, I know he would much prefer to focus on just one topic at a time, but I just never thought of doing a term of this and then a term of that.  Excellent idea. NZ has 4 terms a year, do you think a term of algebra, geo, num theo, counting would work?  It would take him forever to get through that intermediate algebra book.

Yes, that would work nicely. I might even be tempted to cut one of the other subject terms and do two terms of intermediate algebra at this stage. That particular AoPS text really IS huge & chock full of great stuff that a kid like your ds should not miss.

 

Is he done with the Intro Geom text yet? I would finish that off; when it's done, my personal feeling is that the best way to progress in Olympiad geometry is just to work lots & lots of intermediate level problems. Then, in a few years when he's more experienced, have him take Richard Rusczyk's Olympiad Geometry course online. My kids both benefited immensely from his class, but they needed some mathematical maturity first.

 

 

I've been looking at all the number theory books that you and mathwonk have suggested, and I will have to say that the upper undergrad rating on most of them is scaring me off.  Is that really where my child is at?!?!?! Or are these books set up like Zeitz? Where you can work up to your level and then move to the next chapter, studying the book in many passes. The AoPS intermediate course looks good, and the timing is pretty good too.  So I might ask him if he would like to go that route.  He can't do the EPGY classes apparently without taking some sort of test, none of which look to be easily accessible to us.  sigh.

Well, I think that the Stark text is very well written for high schoolers & would be accessible to your ds now. EPGY's class adds in plenty of practice problems and homework sets, which would take you off the hook for prepping! I bet that if you called and explained your situation, that they might allow him to sign up? It's worth a try anyway.

 

MathPath (when dd attended) had a number theory course for middle schoolers like your ds which was taught out of the beginning of Hardy & Wright. They recommended that any kids who were interested in learning further buy that text.

 

We went with the AoPS intermediate course online and the EPGY number theory course for dd instead. Too much math, too little time!

 

 

As for the camp, apparently of the 26 kids going to the camp, they choose 12 to make up the squad.  The squad is then given intensive training.  I think ds is too young for the squad, but the deputy leader did say he could mentor ds starting in September.  I just am not sure what 'mentor' means exactly.  Handing ds books and problems to do would be awesome.  So here's to hoping!

That's not too different from the US training camp (called MOSP) They divide the kids into levels based on testing before arrival. Roughly it's 12 in the top (black) group, another 12-18 in the intermediate (blue) group, and 12-24 in the beginner 9th grade (red) group. The team of six from the US is chosen from the 12 black MOSP kids. My ds was a blue MOSPer in his best year. Even though he didn't make the team, he still received great training at camp and came home with tons of learning materials and problem sets. It's amazing what they can cram into three weeks or so of intense time together.

 

Again, good luck to your son. I'm hoping, too!

[lewelma]

Kathy, your suggestions are really resounding with me.  So I've taken your idea of intensive study and tried to plan out the next 18 months. We have 4 10-week terms per year and we are currently in Term 3 (I've labelled the first and second half of the terms as a and b).

 

Term 3a: exam

Terms 3b & 4:  Finish Geometry text and 3 chapters of Probability in the Counting book

Summer and Term 1a 2014:  AoPS Intermediate Number Theory class  (December to March)

Term1b, 2, & 3 - AoPS Intermediate Algebra

Term 4 - Intermediate Counting

 

 

 

We went with the AoPS intermediate course online and the EPGY number theory course for dd instead. Too much math, too little time!

 

I'm not clear with which route you took.  I think you missed a word.  :001_smile: Also, how is the EPGY class different from the AoPS one?

Even though the AoPS class looks to fit well, I will still contact EPGY and see if they take kids without tests.  Perhaps a portfolio? Because they seem to have a lot of other good classes!   And I will buy the Stark book and Hardy & White for my parents to bring in April, as a follow on to the AoPS class. 

 

 

 

my personal feeling is that the best way to progress in Olympiad geometry is just to work lots & lots of intermediate level problems. Then, in a few years when he's more experienced, have him take Richard Rusczyk's Olympiad Geometry course online. My kids both benefited immensely from his class, but they needed some mathematical maturity first.

  This is excellent advice. Where do I get intermediate level problems?

 

 

 

That's not too different from the US training camp (called MOSP) They divide the kids into levels based on testing before arrival. Roughly it's 12 in the top (black) group, another 12-18 in the intermediate (blue) group, and 12-24 in the beginner 9th grade (red) group. The team of six from the US is chosen from the 12 black MOSP kids. My ds was a blue MOSPer in his best year. Even though he didn't make the team, he still received great training at camp and came home with tons of learning materials and problem sets. It's amazing what they can cram into three weeks or so of intense time together.

  I had no idea that your ds made it to MOSP!  wow!  Because high school starts in 8th grade here, the summer camp allows for 8th graders, although I am guessing that not many apply because it would require the middle school teachers to be aware of the program because the student would have to apply mid-7th grade, and many 7th graders are still in an extended primary here with a single teacher for all subjects.  ds has to take the same exam as the rising 9th and 10th graders.  But here is where the subjective nature comes in, we have been told by the deputy leader that they want to get the kids as early as possible to 'bring them up' so to speak, and they will have to evaluate his performance compared to kids 1 and 2 years older to make a determination about camp attendance.  I'm guessing that it would be a very tough call.  They do not give out scores for this very reason because my ds could have an overall score lower than older kids but get in because they want younger kids. So I think that he is either in or not, with no clarification besides that of how well he did.  Could be very frustrating. They do publish the model solutions, including different approaches so he can learn from his mistakes.

 

So far the whole process has been very very good for ds.  He has loved having a reason for all his study.  Thanks so much for suggesting this path back in November when I realized that we were putting all our time, energy, and money into violin because we had passionate mentors encouraging it. He has been so happy with the switch in focus, even though he is still taking violin with the Concert Master of the NZ Symphony Orchestra!  Now I am a bit more confident talking to his violin tutor and telling him what ds has time for and what he does not have time for.  My role seems to be one of grand coordinator for this child.

 

Ruth in NZ

[Kathy in Richmond]

Kathy, your suggestions are really resounding with me.  So I've taken your idea of intensive study and tried to plan out the next 18 months. We have 4 10-week terms per year and we are currently in Term 3 (I've labelled the first and second half of the terms as a and B).

 

Term 3a: exam

Terms 3b & 4:  Finish Geometry text and 3 chapters of Probability in the Counting book

Summer and Term 1a 2014:  AoPS Intermediate Number Theory class  (December to March)

Term1b, 2, & 3 - AoPS Intermediate Algebra

Term 4 - Intermediate Counting

 

Looks terrific!  

 

 

I'm not clear with which route you took.  I think you missed a word.  :001_smile: Also, how is the EPGY class different from the AoPS one? Even though the AoPS class looks to fit well, I will still contact EPGY and see if they take kids without tests.  Perhaps a portfolio? Because they seem to have a lot of other good classes!   And I will buy the Stark book and Hardy & White for my parents to bring in April, as a follow on to the AoPS class.

 

We used the AoPS intermediate number theory online course first, followed later by EPGY university number theory the next year. The AoPS course doesn't have a textbook or graded problem sets like their other online classes, so there was still room for more learning afterwards. And the EPGY class is longer (a semester as opposed to 8 weeks for AoPS) and touched on more advanced topics. There were daily practice problems and weekly graded problem sets & occasional exams. It did not seem too easy or redundant after the AoPS course.

 

 

This is excellent advice. Where do I get intermediate level problems?

 

 We got our intermediate geometry practice problems from old AIME and USAMTS exams, and from past HMMT and PUMaC problems...basically anything that we could find online that had solutions provided!

 

 

I had no idea that your ds made it to MOSP!  wow!  Because high school starts in 8th grade here, the summer camp allows for 8th graders, although I am guessing that not many apply because it would require the middle school teachers to be aware of the program because the student would have to apply mid-7th grade, and many 7th graders are still in an extended primary here with a single teacher for all subjects.  ds has to take the same exam as the rising 9th and 10th graders.  But here is where the subjective nature comes in, we have been told by the deputy leader that they want to get the kids as early as possible to 'bring them up' so to speak, and they will have to evaluate his performance compared to kids 1 and 2 years older to make a determination about camp attendance.  I'm guessing that it would be a very tough call.  They do not give out scores for this very reason because my ds could have an overall score lower than older kids but get in because they want younger kids. So I think that he is either in or not, with no clarification besides that of how well he did.  Could be very frustrating. They do publish the model solutions, including different approaches so he can learn from his mistakes.

 

So far the whole process has been very very good for ds.  He has loved having a reason for all his study.  Thanks so much for suggesting this path back in November when I realized that we were putting all our time, energy, and money into violin because we had passionate mentors encouraging it. He has been so happy with the switch in focus, even though he is still taking violin with the Concert Master of the NZ Symphony Orchestra!  Now I am a bit more confident talking to his violin tutor and telling him what ds has time for and what he does not have time for.  My role seems to be one of grand coordinator for this child.

My son surprised me when he started doing well in all these contests. I knew he was smart, but we hadn't gone so fast through the math curriculum as others. He's not a competitive type of kid at all, either. He'd been in public school through grade 5, which in itself limited his progress. But he was always deep and careful and persistent in his thinking. The way you describe your son has always made me think that you have a kindred spirit.

 

The olympiad camps love grabbing young kids for training here, too. The ninth grade MOSP program is for that purpose, allowing younger students to come to camp with lower overall scores than the older high schoolers. Only here, they're fairly open about camp score cutoffs. The USACO computer camp is similar, & my ds was on the losing side of that one! The training camp cutoff hit just above his score, except for a couple of super bright younger kids with slightly lower preliminary results. It just makes sense to grab a young student who can be trained over several years instead of an older student with only one year left.

 

Wow on your son's violin training, too. Good luck to him and to you in figuring out how to balance it all! Gotta run now...(at my Mom's visiting with a houseful of company for a few days) :001_smile:

[lewelma]

Thanks for the extra info, Kathy.  Enjoy your family time!

[mathwonk]

when i was a 12 year old in the 8th grade a substitute teacher showed us the most important infinite series there is, and I have never forgotten it.

 

consider S = 1 + r + r^2 + r^3 + r^4+....., and multiply by r:

 

rS = r + r^2 +r^3 +......, and subtract:

 

S - rS = 1, and solve for S = 1/(1-r) = 1+r +r^2 +r^3 +........

 

 

This symbolic manipulation actually makes sense for |r| < 1.

 

E.g. 1 + 1/3 + 1/3^2 + 1/3^3 +.....= 1/(1-1/3) = 3/2.

 

 

 

This is the most important infinite series in the world, and has many uses.

 

And virtually any child can follow this derivation.

 

In particular there is little need to wait for the last chapter of

 

some book or other to learn this.

 

 

e.g. this explains how to evaluate a repeating decimal like .444444...

 

 

(4/10)(1+ 1/10 + 1/10^2 + 1/10^3 +.....) = (4/10)(1/(1-1/10)) = (4/10)(10/9) = 4/9.

 

 

or even easier, redo the derivation: S =.4444..., so 10S = 4.44444...

 

so 10S-S = 9S = 4.444... - .4444.. = 4, so S = 4/9.

 

 

 

Even in calculus, the study of "absolute convergence" by means of basic tests

 

such as the "ratio test" or "root test", just reduces to comparing

 

a given series with this one.

 

 

I.e. the truth is this is the one series we understand best,

 

and so it gets used over and over. Hence I submit this as the minimum

 

knowledge of infinite series everyone might benefit from, as early as possible.

[lewelma]

Thanks Mathwonk.  I have printed your text and will go over it with ds tomorrow to make sure he knows those.  My favorite quote is "chance favours the prepared mind."  The more he is familiar with, the more he will see on the test!

[Arcadia]

A couple of classic number theory texts for his shelf: (very expensive; maybe you can locate a used copy)

 

Niven, Zuckerman, & Montgomery, An Introduction to the Theory of Numbers

Hardy & Wright, An Introduction to the Theory of Numbers

Thanks. Found the PDF versions for both books.

The Hardy & Wright book is on archive.org https://ia800201.us.archive.org/23/items/AnIntroductionToTheTheoryOfNumbers-4thEd-G.h.HardyE.m.Wright/AnIntroductionToTheTheoryOfNumbers-4thEd-G.h.HardyE.m.Wright.pdf

[CadenceSophia]

Thanks. Found the PDF versions for both books.

The Hardy & Wright book is on archive.org https://ia800201.us.archive.org/23/items/AnIntroductionToTheTheoryOfNumbers-4thEd-G.h.HardyE.m.Wright/AnIntroductionToTheTheoryOfNumbers-4thEd-G.h.HardyE.m.Wright.pdf

To each their ethics, but nearly any text book can be found on Library Genesis.