Expanding the Wiki - Number Theory - Calvin Lin

$\color{#D61F06} { \text{Updated! October 2} }$

The Number Theory Section has now been published.

You will notice that there are skills without a star, and only has the wiki book icon. This indicates that there are no practice problems yet, though you can still read the writeups.

If you want to extend the (currently published) chapters by submitting writeups on specific skills that are not published, please let me know.

$\color{#D61F06} { \text{Updated! September 30} }$

I have updated the document with the list of suggestions, and sorted the skills out into corresponding chapters and subtopics. If you would like to volunteer a writeup for any of these skills, please reply below to the comment with the corresponding topic name to let me know. I will record down your name on the spreadsheet.

Priority of publication is given to skills which are more likely to have activity on them. I have shaded some items in red, to indicate that I do not think there will be activity on them. However, if the community is willing to contribute articles in those chapters, then I will publish those skills accordingly.

I would like to get this out as quickly as possible, and look forward to reading your contributions.

I am interested in expanding the Wiki further beyond the current Practice section.

For now, the easiest approach would be to build out more skills that are applicable to solving the problems. As such, I would like to get started on the Number Theory Map. We have given some thought to how it should look like. However, the list of skills is still very minimal, and I would appreciate additional input. You can view (but not edit) the current list in the Number Theory tab on Upcoming Skill Maps.

Please leave feedback (in this note) about what skills you think should be included, to make the Number Theory map comprehensive and useful for other beginner Number Theorists. Feel free to suggest relevant skills, or alternative ways of organizing the material.

Note: I will be working on Olympiad Algebra over the next few weeks. Comments are also welcome, though I will be making another general announcement when I get started on it.

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Math

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Remember to wrap math in $ ... $ or \[ ... \] to ensure proper formatting.

2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

\sqrt{2}

\sum_{i=1}^3

\sum_{i=1}^3

\sin \theta

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\boxed{123}

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Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

"I want to add a skill to an existing chapter in Number Theory"

Reply to this comment if you want to contribute a writeup to a skill, but it has not been published yet. Please state the title of the skill, and the chapter that it would fit under.

Calvin Lin Staff - 6 years, 8 months ago

Integers

Reply to this comment if you want to contribute an article

Reply to this comment if you want to add a specific skill / chapter to this subtopic.

I would like to contribute articles on Number Bases and Factors.

Pranshu Gaba - 6 years, 8 months ago

Can you state the specific skills? This would help other members identify the remaining skills. Thanks!

@Calvin Lin – In Factors, I have already written an article on number, sum and product of factors. (How can I submit this?). In Number bases, I am interested in writing about Binary.

@Pranshu Gaba – Thanks, I have recorded your name down in the document. I will inform you when the Wiki pages are up, and you can add them directly.

Thanks for your assistance!

@Calvin Lin – It's my pleasure!! Looking forward to contribute more. :D

@Pranshu Gaba – @Pranshu Gaba @Brian Kal This is now live. You can refer to it in Number Theory - Integers.

I like

Ubaid Kc - 6 years, 8 months ago

I can do more integer-related stuff. If you've seen (and liked) my post on the nugget theorem, I can create more articles related to that, if you wish

Brian Kal - 6 years, 8 months ago

Fractions

Modular Arithmetic

I would be happy to contribute an article (or at least part of an article) on modular arithmetic. In particular think Pythagorean triples would combine with modular arithmetic very nicely. Also, I can provide detailed proofs of how certain irrational numbers are derived and methods of approximating surds (i.e. Pells equation ).

Curtis Clement - 6 years, 5 months ago

That's great! I'm excited that the community is eager to contribute to the wikis.

I would appreciate a more detailed explanation / examples for Chinese Remainder Theorem and Finding last digit of a number. If you're feeling particularly adventurous, writeups for Pell's equation or Lucas' Theorem would be lovely :)

Calvin Lin Staff - 6 years, 5 months ago

@Calvin Lin – How can I add these articles. Also, how to I type the equivalent/triples equal sign for modular arithmetic

@Curtis Clement – For those topics that I mentioned, if you click on the links you will land on the wiki pages.

If you wish to create a new wiki page, select "Post something" from the top header, and select "Wiki" after that.

\equiv gives us $\equiv$ . Note: If you see any latex symbols around, you can hover your mouse over it and the latex code will be displayed.

Diophantine Equations

I have already said about your Blog. Or You are not interested? http://www.artofproblemsolving.com/Forum/blog.php?u=206450&

Andrey Loschinin - 6 years, 8 months ago

I would like to write about Mordell's equation.

Bogdan Simeonov - 6 years, 8 months ago

I would also like to contribute in Diophantine equations

lawrence Bush - 6 years, 8 months ago

Certainly! I've added your name to the chapter. Are there any specific skills that you are interested in?

No,generally all diophantines will do.Thanks for such a fast reply.

@lawrence Bush @Bogdan Simeonov @Andrey Loschinin

This is now live! You can refer to it in Number Theory - Diophantine Equations

How do you add in notes that you've already saved to your profile? Or should I just copy/paste the info from my note into a new note for the wiki page?

@Brian Kal – Copy and paste is great!

Ummm...couldn't find anything about Mordell's equation

Arithmetic Functions

I would like to contribute for an article.

@Calvin Lin – Generally about convolutions,multiplicative functions,their partial sums(a tiny bit of analytic number theory).

@Bogdan Simeonov – How exactly do I submit?

@Bogdan Simeonov – Thanks. I've added your name to the document for the relevant skills. Once I publish the maps, I will let you know and you can add your writeup to the Wiki page directly, and be recognized as an editor. Thanks!

Algebraic Number Theory

You can add Euler's Totient Function and Riemann Zeta's Function.

Yash Singhal - 6 years, 8 months ago

Riemann zeta function is a good addition. Would you like to do a write up on it?

The totient function has already been covered under eulers theorem. I don't think it deserves a separate skill.

@Calvin Lin – Yes, I would like to write it. BTW, I have written some writeups, you can add my name in your Upcoming Skills Document. I have written on the following topics:

Prime Numbers
Distribution of Primes
Infinitely many Primes
GCD and LCM
Even and Odd numbers
Binary System
Rule of Product and Rule of Sum(Combinatorics).

I have seen some topics in red colour of your Upcoming Skills Document like Goldbach, Twin Prime Conjecture. You have not launched it as you require writeups. I am ready to write about them but I request you to add them in our practise section. Thanks!!!

You can also add the Tau Function.

Number Theory Applications

There are a lot of topics in number theory, but I imagine you'd like to limit them to those of "practical use", i.e. can be of help in solving problems in elementary mathematics, instead of, for example, some theoretical topics in algebraic number theory. As a matter of fact, does Brilliant plan on discussing group theory? There is a strong connection between group theory and number theory, as well as geometry and algebra.

Michael Mendrin - 6 years, 8 months ago

The Wiki will not be limited to "must be useful in problem solving". There are many concepts that I can think of that should be in the wiki, that are not of "practical use". For example, I think that the Collatz Conjecture and Goldbach's Conjecture are worthy of a writeup. (And if someone wants to tackle an explanation of Zhang Yitang's proof, I would love to see it.) Theoretical topics are also welcome, if there are contributions.

Edit: I have started marking such skills in Blue in the document.

I would like to touch group theory. I did post some notes on them a while back. I will likely place it in the "Olympiad Algebra" section, especially if there is interest from the community. I will drop you a note when I plan that section.

Edit: I have added Group Theory and Elliptic Curves to the map. You are welcome to suggest skills in these chapters.

Sir, I have no prior experience but, I would ask whether you would add Hansel's lemma (or, LTE),Zsigmondy's and such elementary NT topics. I would like to use the Wiki to master, so, I may inquire whether these will abode in the Wiki or not. And, I beleive, under the chapter Pell's equation, there may lye a skill called Quadratic Surds. I am not a fact-man (I am practicing to be one), so, please ensure me.

Sheikh Asif Imran Shouborno - 6 years, 8 months ago

Thanks, those are very good ideas. I've added Hansel's Lemma and Zsigmondy's theorem to the list.

Yes, these will have a corresponding Wiki page, which the community could add their ideas to.

Pells Equation involves looking at quadratic surds. To solve the Pell's equation $x^2 - D y^2 = 1$ , we look at the continued fraction representation of $\sqrt{D}$ .

Good thoughts.

But hey! Quadratic surd in Pell's equation?

Jubayer Nirjhor - 6 years, 8 months ago

Yes, @Jubayer Nirjhor . You can see, the book by Edward Barbeau includes a chapter on quadratic surds. I don't have a hard copy of the book; I've just gazed at it. I hope the Wiki helps us. By the way, we always need to use hard copy books. Wiki can be a nice supplement. ....And, sometimes, completely unusual stuffs that motivates a learner can be included in a Wiki. What do you think?

I am planning to advent such doc section in our math club group in Facebook, with the help of you all, that keeps the pace on the move and keeps us abreast of the world. I think this Wiki can turn out to be a nice reference resource!

really Calvin Lin you doing gr8 job,

gopalgoel goel - 6 years, 8 months ago

Wiki project you have undertaken is very huge project. I congratulate and thank you for that. I think Wiki is for the beginners as well for those who would use it as reference.

I feel, it will be easy to set standards now itself. To make correction afterwards would become difficult. So, to have uniformity to make things easy for beginners, there should be uniform notations. The variables used also must be same as far as possible throughout. Say N = r (mod m). Variables used for prime number would be from only certain letters, same for integers………. It will be time consuming to set standards, but I think it will be rewording labor. I am sorry I do not know much of the subject to contribute. For this the author can not put the article directly, but you have to edit it.

Niranjan Khanderia - 6 years, 8 months ago

Thanks for your encouragement. I'm excited that the Brilliant community is so responsive to the Wiki, as it indicates that this a need that many people have.

We are working on setting the standard of the page, and are reviewing feedback from members of the community. For example, some of them have requested for more examples, since that is the way they prefer to learn. As we gain more information, we will update it in Guidelines for a good Wiki editor. I agree that setting a standard for the quality of the writeups would be rewarding.

I am less inclined to conforming variables that are used, in part because there is no such consensus across the community, and Mathematics is "independent" of the variables.

Thanks. I was thinking about fresh students who would be a little confused when they have learned a theorem with triangle as ABC when they have to prove for triangle LMN!!! Not a question of standardized name but of convenience for the learner. May not be important for us in Brilliant.

I was thinking if odd and even numbers were explained before primes,it would help in the definition of primes. This is,if the order will be like what you have posted in the spreadsheet.

Adarsh Kumar - 6 years, 8 months ago

The order of the skills is not fixed, and I certainly welcome suggestions about how to reorder them.

To me, Prime numbers are a more basic part of integers, though I agree that even and odd numbers are much simpler to explain and understand. I will give more thought to how those should be arranged.

Ok.

Thanks......................................Converting
$( m+\sqrt n)~~ to~~ (p+\sqrt q)^2 ~is~OK,~~ but~ how~ to~~ convert~ into~ cube~ if~ it~ is~possible. \\ 1-\sqrt2 -\sqrt3 +\sqrt6 = \sqrt{ a+ \sqrt {b+ \sqrt{ c+\sqrt{ d}}}} ~~~a,b,c,d??? \\1-\sqrt2 -\sqrt3 +\sqrt6 = (1-\sqrt2)*(1 -\sqrt3 )~~I~ know~that ~much.$

I tried to search the web for deeper manipulation of surds, but could not find much. In irrational numbers can you write on this ?

Can you clarify what you mean by "deeper manipulation of surds"?

Can you suggest some specific areas of irrational numbers that you are interested in?

Aerchimedian property of real numbers

vishal ginda - 6 years, 8 months ago

Thanks! I forgot about that.

you are so coooll

fatimah latif - 6 years, 8 months ago

I know very little about Number Theory . Probably , Finite and Infinite continued SURDs may be added. Some thing more on surd beyond elementary manipulation.,

Sir, I can contribute the articles which will be very helpful for those who are preparing for JEE. But I don't have any idea about how to contribute an article. And what that article should include ? Actually, I am unknown about wiki. I will try my best !

Sandeep Bhardwaj - 6 years, 8 months ago

@Calvin Lin Sir, I would be greatly interested in contributing towards inequalities section. Also, Cant we just add some more to inequalities part, like Stolarsky's theorem, Majorization, etc. Those methods helps to kill a lot of problem. I would be really pleased to contribute towards the inequalities tab.

Dinesh Chavan - 6 years, 8 months ago

Thanks for your feedback. Once I launch the Number Theory section, I will be working on extending the Algebra section, which will include inequalities. You can look at the Upcoming Skill Maps document.

I am not much competent to contribute any full articles. But I have a few points. I am dividing these into different comments. Our Brilliant family has an atmosphere of cooperation and helping and no rivalry. This is very helpful for all of us.

It is very good that you answer us even if there is a small matter. But I feel there should be some one else to do such trivial things so that your time is utilized most efficiently.

. Indexing similar to that for library books can be done for the problems. We have sections Number Theory, Algebra…….we can have subsections and number problems accordingly. We can start with indexing and numbering with new problem and then slowly go into the old ones.

how do I comment in your spreadsheet?

You can comment here. The ideas will be taken into consideration

Agnishom Chattopadhyay - 6 years, 8 months ago

As mentioned, "You can view (but not edit) the current list ... Please leave feedback (in this note) ".

The attached document is intended to reflect a "finalized" version, which is why I didn't allow members to edit as yet. Moderators have the ability to edit the document, as they have been responsible for determining the skill tree.

Note: I have removed some of the comments, to keep this discussion focused on the map.

Yes sir,I got it.Thanx for replying.

Hello. In my Blog you can see the solutions of some Diophantine equations. If you want to place, then give a link to my Blog. Formula I brought all. http://www.artofproblemsolving.com/Forum/blog.php?u=206450

Thanks for your suggestion. You are welcome to suggest skills under which your writeups would fit, especially since the Diophantine Equations Chapter isn't well developed yet.

At this stage, we are looking for direct contributions, as opposed to merely linking out of the site into other places.

Mental maths and IQ should be included.

Sumant Chaudhary - 6 years, 8 months ago

Thanks for your suggestion. I have added it to the "General" tab for now, as it is not directly relevant to Number Theory.

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$