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Old 2015-10-26, 14:35   #1
Xyzzy
 
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Cool "A First Course in Number Theory" discussion group

Mr. Silverman has recommended this book for us to read to learn the basics of number theory.

http://www.amazon.com/First-Course-N.../dp/0534085148

We have ordered a copy. It looks like you can get one for around $15 shipped easily.

The idea we propose is that maybe some of you all will get the same book and then as a forum we can (slowly!) work through the book. (Buy your book now before the demand drives the prices up!)

It would be awesome if people who are already proficient in this field would be available to help.

What do you all think of this idea?

We picture this as a long-term project, similar to one of the forum's chess games. If this idea takes off we can make a separate subforum for it, or whatever you all want to do.

Thoughts and suggestions are appreciated!

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Old 2015-10-26, 18:35   #2
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Quote:
Originally Posted by Xyzzy View Post
What do you all think of this idea?

We picture this as a long-term project, similar to one of the forum's chess games. If this idea takes off we can make a separate subforum for it, or whatever you all want to do.

Thoughts and suggestions are appreciated!
This will be a good place to point people to as well.

What other math courses should one have taken (or learned) as a pre-requisite for this?
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Old 2015-10-26, 18:43   #3
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Originally Posted by Uncwilly View Post
This will be a good place to point people to as well.

What other math courses should one have taken (or learned) as a pre-requisite for this?
Good knowledge of high school level algebra and the ability to apply a sequence of reasoning and
do proofs.
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Old 2015-10-26, 19:35   #4
xilman
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Originally Posted by R.D. Silverman View Post
Good knowledge of high school level algebra and the ability to apply a sequence of reasoning and do proofs.
I'm not convinced that being able to create rigorous proofs is absolutely essential. Highly desirable, of course, as is the ability to read and understand proofs given by others.

Much valuable mathematics has been inspired without rigorous proofs being produced. The Riemann hypothesis is a notorious example, as is Fermat's last theorem. Many results in computational number theory are conditional on unproven conjectures.
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Old 2015-10-26, 19:44   #5
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Quote:
Originally Posted by R.D. Silverman View Post
Good knowledge of high school level algebra and the ability to apply a sequence of reasoning and do proofs.
On a more constructive note than my previous comment --- the ability to perform background reading, including knowing how to search for background information and how to ask questions should it prove necessary afterwards. In summary: to understand the difference between training and education.
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Old 2015-10-26, 21:16   #6
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I'm not convinced that being able to create rigorous proofs is absolutely essential. Highly desirable, of course, as is the ability to read and understand proofs given by others.
One needs to be able to take a statement and show that it is true. And why it is true....

e.g. as a start, derive the Fundamental Theorem of Arithmetic. Show that it is true.

Do people need a hint? Start by Proving: If p is prime and p | ab then p|a or p|b.
Note that this is NOT true if p isn't prime, e.g. 6 | (4*3) but 6 divides neither 4 nor 3.


Unfortunately, it is too easy for people to cheat (and they cheat themselves in the process) by
looking such things up on Wiki.
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Old 2015-10-26, 21:32   #7
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Quote:
Originally Posted by R.D. Silverman View Post
One needs to be able to take a statement and show that it is true. And why it is true....

e.g. as a start, derive the Fundamental Theorem of Arithmetic. Show that it is true.
Assume I'm ignorant.

Please edify me as to how I would do that.

Sincerely.
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Old 2015-10-26, 21:33   #8
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Quote:
Originally Posted by R.D. Silverman View Post
One needs to be able to take a statement and show that it is true. And why it is true....

e.g. as a start, derive the Fundamental Theorem of Arithmetic. Show that it is true.

Do people need a hint? Start by Proving: If p is prime and p | ab then p|a or p|b.
Note that this is NOT true if p isn't prime, e.g. 6 | (4*3) but 6 divides neither 4 nor 3.


Unfortunately, it is too easy for people to cheat (and they cheat themselves in the process) by
looking such things up on Wiki.
Sure.

I'm trying to draw a distinction between proving the FTA and, say, proving that the NFS runs in L(1/3) time.
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Old 2015-10-26, 21:52   #9
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Quote:
Originally Posted by Xyzzy View Post
Mr. Silverman has recommended this book for us to read to learn the basics of number theory.

http://www.amazon.com/First-Course-N.../dp/0534085148

We have ordered a copy. It looks like you can get one for around $15 shipped easily.

The idea we propose is that maybe some of you all will get the same book and then as a forum we can (slowly!) work through the book. (Buy your book now before the demand drives the prices up!)

It would be awesome if people who are already proficient in this field would be available to help.

What do you all think of this idea?

We picture this as a long-term project, similar to one of the forum's chess games. If this idea takes off we can make a separate subforum for it, or whatever you all want to do.

Thoughts and suggestions are appreciated!

I've just ordered a copy; consider me onboard. Although I already have a few number theory books, I've acquired them much like a disorganized person buys organizers multiple times but never particularly knuckles down.

A common path among acquaintances with the benevolence of slight mutual mentoring sounds nice.
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Old 2015-10-26, 22:23   #10
R.D. Silverman
 
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Originally Posted by blip View Post
I am coming back to my previous recommendation:

Henri Cohen, A course in computational algebraic number theory, Springer GTM 138

As can be seen from the contents, this is just spot-on the material we tend to discuss here.

Google is your friend...
NO! Much too advanced. It assumes that you have already taken a course in
linear algebra, modern algebra, and number theory.

Do you know what it means for a group to act on a set by conjugation?
no? Then you are not ready for this book.
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Old 2015-10-26, 22:31   #11
R.D. Silverman
 
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Originally Posted by chalsall View Post
Assume I'm ignorant.

Please edify me as to how I would do that.

Sincerely.
Are you here to learn, or are you here simply to be given the answers to questions?
Or is this just a taunt?

As with all homework (not just math!) one learns by doing.

Do you know what the theorem says? Look it up. Now ask:

What do I need to show in order to establish that the result is true?

This is one of the most basic and fundamental results in number theory.

If the theorem as a whole is too hard, can you prove the following,

If p,q are primes, then p*q != r*s, where r,s are not equal to either p or q.
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