Four squares theorem and Mathematica

How can we represent, say, 123456 as a sum of four squares? Can it be done in more than one way, perhaps?

Yes, it can be done in exactly 181 ways. Three examples are:
$123456 = 0^2+8^2+176^2+304^2$
$123456 =28^2+172^2+172^2+252^2$
$123456 =4^2+12^2+236^2+260^2$

Representations like this can be calculated with Mathematica, use the PowersRepresentations function.

Calculations like this are expensive, i.e. can take a long time. Could be interesting to have study the algorithm.

Sofar, I read the proof of Lagrange's four-square theorem in three books. ( As a preparation for the general proof of the polygonal number theorem by Cauchy. ) Although they all (i.e. Nathanson, Burton and Davenport, ) use the same proof the clarity differs greatly among these authors. At a certain level too much verbosity doesn't add to the clarity anymore but nothing is worse than too much density. Only Davenport used an example to illustrate the proof, thanks to his text I am beginning to understand the proof.

Calculus video lectures at the Worldwide Center of Mathematics

The Worldwide Center of Mathematics publishes ( modern... ) calculus textbooks ( not a bad idea if you ask James Stewart ) and has a section with freely available video lectures on calculus and multi-variable calculus as well as a section with talks on the research level. There are also talks about research which are meant for undergraduates. Sort of to give you an idea of the field these researchers operate in.

Well, the Calculus lectures can be of help if you are on MST121, MS221 or MST209. There are a lot of options nowadays if you are looking for calculus videos.

Complex Analysis for Number Theory

I got hold of a book from Anatoly A. Karatsuba with the title Complex Analysis for Number Theory.

Chapter 1. The Complex Integration Method and Its Application in Number Theory.
Chapter 2. Riemann Zeta
Chapter 3. Dirichlet L-functions

Paragraph 1. Generating Functions in Number Theory. Section 1. Dirichlet Series. Can it be more to the point? This book is clearly written with the intention of efficiently transferring the required tools of complex analysis to number theory students. Chapter 2 introduces the Riemann Zeta function.

Why is this book priced at $179.95 at Amazon? Because it is hard-cover? It is a book from the 19-nineties. This is educational material. Transfer it to electronic format and price it reasonably.

What is... additive number theory ?

The central problem in additive number theory is to determine if a given set of integers is a basis of finite order.

Take for example the square numbers. According to Lagrange's theorem every integer can be represented as a sum of at most four squares. ( The number 7 for example, requires four squares: 7 = 2^2 + 1^2 + 1^2 + 1^2. ) In this example the set of squares is a basis of order 4.

Goldbach's conjecture is a famous problem in additive number theory.

Fermat's polygonal number theorem

Fermat

Imo propositionem pulcherrimam et maxime generalem nos primi deteximus:nempe omnem numerum vel esse triangulum vex ex duobus aut tribus triangulis compositum: esse quadratum vel ex duobus auttribus aut quatuorquadratis compositum: esse pentagonum vel ex duobus,tribus, quatuor aut quinque pentagonis compositum; et sic deinceps in infinitum, in hexagonis, heptagonis polygonis quibuslibet, enuntianda videlicet pro numero angulorum generali et mirabili propostione. Ejus autem demonstrationem, quae ex multis variis et abstrusissimis numerorum mysteriis derivatur, hic apponere non licet....

I have discovered a most beautiful theorem of the greatest generality: Every number is a triangular number or the sum of two or three triangular numbers; every number is a square or the sum of two, three, or four squares; every number is a pentagonal number or the sum of two, three, four, or five pentagonal numbers; and so on for hexagonal numbers, heptagonal numbers, and all other polygonal numbers. The precise statement of this very beautiful and general theorem depends on the number of the angles. The theorem is based on the most diverse and abstruse mysteries of numbers, but I am not able to include the proof here....

Pierre de Fermat, +/-1650

As with his famous Last Theorem, Fermat had no proof. Gauss proved the case for triangles, Lagrange for squares and Cauchy finally proved the general case. Apostol wrote in his book Analytical Number Theory:

For example, Fermat proved the following surprising theorems: Every integer is either a triangular number or a sum of 2 or 3 triangular numbers; every integer is either a square or a sum of 2, 3, or 4 squares; every integer is either a pentagonal number or the sum of 2, 3, 4, or 5 pentagonal numbers, and so on.

Tom Apostol, 1976

Perhaps Apostol did not know the correct history of the polygonol number theorem. Remember that there was no internet in 1976.

I wanted to see that proof but Apostol had not provided one for obvious reasons. The M381 book did not gave a proof either, so that didn't help much. I checked W'pedia: no proof. Mathworld: no proof. In cases like that there is always Planet Math to the rescue. Not this time anyway. I tried entering a few queries through the OU Library Service, but they returned too many hits. So that did not work either. Anyway, I finally found a hint at the site called 'Fermat polygonal number theorem'. The book Additive Number Theory - The Classical Bases by M. Nathanson has a chapter devoted to it.

I practically beamed myself to the library and picked up a copy. So, I am about to study this enigmatic theorem. Even if I am not ready for this proof yet, I can pinpoint topics for further study. - Chapter 13 'Representation of Integers as Sums of Squares' in Elementary Number Theory by David Burton has Lagrange's proof as well.

Perfect numbers

An example of a note in my NT Wiki. In M381 only the first part of the proof is given which was known as early as Euclid. Euler was the first who gave, a not so very clear proof of the reverse. Many followed Euler with subsequent improvements of the proof. I have used a proof of Dickson published in 1911. - Perfect numbers are still actively researched. It is for example still unknown if odd perfect numbers exist. If they exist however they are surely very large.

An even integer is perfect if and only if it can be written as $2^{p-1}(2^p-1)$, where both $p$ and $2^p-1$ are prime.

We show that:
If $n = 2^{p-1}(2^p-1)$ and $p$ and $2^p-1$ are prime then $n$ is perfect.
\begin{align}
\sigma(n) &= \sigma(2^{p-1}(2^p-1)) \\
&= \sigma(2^{p-1}) \sigma(2^p-1) \ \\
& = ( 2^p-1 ) 2^p \\
& = 2^p ( 2^p-1 ) \\
& = 2 (2^{p-1}(2^p-1)) \\
& = 2n
\end{align}
Since $\sigma(n)=2n$ we conclude that $n$ is perfect.

We show that:
If $n$ is even and perfect then it can be represented as $n=2^{p-1}(2^p-1)$ where $p$ and $2^p-1$ are prime.

Since $n$ is even we assume $n=2^{k-1}m$ where $(2^{k-1}, m) = 1$.
(1) We calculate the divisor-sum of $n$ as follows:
\begin{align}
\sigma(n) &= \sigma(2^{k-1}m)\\
& = \sigma(2^{k-1})\sigma(m)\\
& = \frac{2^k-1}{2-1}\sigma(n).
\end{align}

(2) We assume $n$ is perfect thus:
\begin{align}
\sigma(n) &= 2n\\
& = 2 (2^{k-1} m)\\
& = 2^k m
\end{align}

Now (1) and (2) gives:
\begin{align}
\frac{2^k-1}{2-1}\sigma(m) & = 2^k m \Leftrightarrow \\
\sigma(m) & = \frac{2^k m}{2^k-1} \\
& = \frac{((2^k-1)+1)m}{2^k-1} \\
& = m + \frac{m}{2^k-1}
\end{align}

Since $\sigma(m)$ is the sum of all divisors $\frac{m}{2^k-1}$ must be $1$. So $m=2^k-1$ with divisors $1$ and $m$ and is thus prime.

We conclude that $n=2^{k-1}(2^k-1)$ with $2^k-1$ and $k$ prime.
QED

The concrete tetrahedron

I once wrote that I rated Concrete Mathematics 6 out of 5 stars. Unfortunately the book was published in the pre LaTeX era. To the eyes or 21st century readers the book simply looks ugly. If you look a bit deeper though you'll notice that you struck gold as far as content is concerned. But... what we can do today on a laptop with Mathematica installed on it was beyond the possibilities of the super computers in the days CM was written. In that respect the book looks out-dated.

Recently a book by Kauers, Palle was published by Springer called 'The Concrete Tetrahedron'. In the book the concrete tetrahedron stands for:
- symbolic sums;
- recurrence equations;
- asymptotic estimates;
- generating functions.

The authors have the following to say about it.

... the present book is not meant to be merely a summary of “Concrete Mathematics”. We have a new twist to add to the matter, and this is computer algebra. In the last decade of the 20th century, many algorithms have been discovered by which much of the most tedious and error-prone work about the four vertices of the Concrete Tetrahedron can be performed by simply pressing a button. We believe that a mathematics student of the 21st century must be able to use these algorithms, and so we will devote a great part of this book to explaining what can and should be left to a computer, and what can and should be still better done the traditional way.

In Apostol's Analytic Number Theory formal power series and Dirichlet multiplication are among the topics. For me that was a reason to refresh, review my knowledge on formal power series and generating functions. ( One of my favorite subjects in mathematics. ) That is how I became aware of this new release.

Link: The Concrete Tetrahedron

Link: Video lectures about Concrete Mathematics

A database of sequences

In the Online Encyclopedia of Integer Sequences ( OEIS ) you can find tons of information on any sequence you can possibly think of.

An example.
A002024 n appears n times.
Starts with: 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6, ...
Closed form: a[n] = Floor[1/2 + Sqrt[2n] ]

Amazing.

Some of my favourite tools

A while back in my 'How to learn from failure post' I wrote about monitoring results. That post assumed to much fore knowledge with my readers about Admin Scales and so on. Admin Scales are essential to me, I have considered writing a post about them but the subject is 1) off-topic and 2) often alienates people, which is not my intention.

Planning, execution of plans and recording efforts and results come before the implementation of a system based on statistics ( and ethics ). As an individual you want the benefits of planning and control but not the costs. Investing too much time in planning makes it inefficient in no-time. Most people work with some form of a todo list. Often in digital form managed by an iPhone app or similar. Todo lists and so forth only work under certain conditions as brilliantly discovered by David Allen, the inventor of GTD. Since 'GTD', most personal planning systems are adaptions of his method. Allen's company sells an Outlook add-on as an example of a good technical implementation of GTD. They also support Personal Brain for example.

Admin Scales have a BE, DO and HAVE dimension. For the DO dimension I use MindManager ( have been using it for brainstorming for more than 10 years ) in combination with MyLife Organized ( MLO ). MLO imports MindManager mindmaps, has a desktop version and App versions for iPhone, Blackberry, PocketPC and Android(soon). MLO is NOT a normal todo list program. Once you get adjusted to it, it acts like someone who thinks with you and knows which item of your huge ( hierarchical ) todo list you should do first. It has an outliner view and a do-now view. It is a phenomenal piece of software but it takes time to appreciate all the features.

Working with images in MediaWiki

If you refer to an image-file in an article make sure that you have the correct settings in your LocalSettings.php, like :

- $wgEnableUploads = true; 

And if you want to use images from the WikiMedia Commons

- $wgUseInstantCommons = true; 

Screenshot of the NT Wiki sofar

I am not going to publish the Wiki yet because not only content is being added daily, I am still working on the structure, i.e. categories and templates, standards and guidelines.

Foundation of the natural numbers

At last I found a more in depth explanation of the natural numbers. In this idea numbers are labels assigned to sets and adding 1 to a number represents a map between two sets. If we would analyze the mathematics of a hypothetical alien civilization we would see that they have natural numbers too, they just used different labels in their definition.

Mathematically it roughly works like the following:

Let A be the collection of all sets.

Let f be a map
 f:A->A
 a: |-> a U {a}

Assign label '0' to Emptyset
If a has label 'x' then assign label 'x+1' to f[a]

Label a f[a]  a U {a} 
--------------------- 
0 Empty 

1  f[Empty] Empty U {Empty}
    = {Empty}
    = {0}

2 {0} f[{0}]  {0} U {{0}}
    = {0, {0}}
    = {0, 1}

3 {0,1} f[{0,1}] {0,1} U {{0,1}}
    ={0,1,{0,1}}
    ={0,1,2} 

More later. ( Confirmed as an article in the NT Wiki. )

Art and mathematics: 3D fractals

A 3D fractal from phidelity.com

Rena Jones - Open Me Slowly 3D Fractal Video from Kris Northern on Vimeo.

( Watch it full screen. )

Math 3110 ( number theory video ) lecture 1

Watched Math 3110 ( number theory video ) lecture 1 to get an impression of the series. Will it be worth it watching the series again but this time in the 'Adams edition'? - Well, considering I still have to do a number theory exam in october, I better should.

One of the prerequisites of the course is an understanding of and experience with the following types of mathematical proof:
- direct
- contrapositive
- by contradiction
- by ( mathematical ) induction.

The first lecture starts with 20 min or so admin trivia which is only relevant for students who are actually doing the course at UCCS. The mathematics in this lecture starts with the WOP and ends with a proof by induction of the sum of an arithmetic sequence. See also: five proofs of the sum of 1,2,3, ..., n.

The required textbook for Math 311(0) is Elementary Number Theory by David Burton ( 7th ed. ).

Mathematics, an intellectual instrument. ( David Borton, History of Mathematics )

Two different series of video lectures on Number Theory

Last year I watched the entire series of Math 311 Number Theory ( Spring series 2010 ) by Prof Dr. Seung Son. The course has been renamed to Math3110 The Theory of Numbers and is currently in progress ( Spring series 2011 ) with a new lecturer Prof. Gene Abrams. Currently 10/30 lectures are online, which are more or less covering OU M381 Units 1-4. With the first lecture on the Well-ordening principle. If you understand the WOP then you really -understand- mathematical induction, which is a deeper level of being able to do induction proofs.

Considering that these videos are free to watch ( thank you UCCS Colorado ) my entire view of the world lightens up. In my view of a cleared  planet all our basic needs are cared for ( by robots ) and we homo novis are studying and communicating, mostly on-line.

Creating a fractal with Excel or Calc

If you have Excel 2010 then do the following, otherwise apply the equivalent code / options for your spreadsheet.

Create a new workbook
Make sure that zeros are displayed as whitespace:
- \File\Options
- Select 'Advanced' on left sidebar
- ( Look for 'Display options for this worksheet: Sheet1' )
- When found deselect 'Show a zero in cells that have a zero value'
Set the column-width to the first 130 rows or so to about 3.0

Now apply some values to the following four cells.
A1: p ( any prime ), use a small prime, for example 3.
C3: 1
A4: =A3+1 ( not necessary, but shows interesting properties )
In C4 we implement the Pascal recurrence modulo a prime
C4: =MOD(C3+B3;*A*1) ( replace * with dollar sign conflict with MathJax )
Now you have to do some copying and pasting:
- you have to copy this cell at least 128 cells to the right
- copy the contents of row 4 at least 127 times

If you have done this correct you will see patterns like this:

Mod 2, zoom 30%

Mod 3, zoom 30%

Mod 7, zoom 30%

It will be interesting to see what happens if you try mod p to a power, i.e. 4, 8 and 16, or mod 9.

It should be possible to control graphics at the pixel level with Mathematica, I might give it a try.

It was OU course M381 ( which I am on at the moment ) that gave me the idea to look at the Pascal triangle modulo a prime because one of the topics in M381 is the Fibonacci series modulo a prime. The Pascal Triangle and the Fibonacci series are definitely among my favourite mathematical objects. If I had to choose I would say the PT of course, because the Fibonacci series is neatly contained within it. What is not? One might ask.

How are your basic math skills?

I don't know how the situation is elsewhere but in The Netherlands there has been a lot of commotion about the basic math skills of teachers in primary schools. Since then students have to do an extra exam in basic math skills. The root cause was that basic math skills were only taught at the primary school level. In secondary school they got dependent on their calculator.

Link: Basic Math Skills Quiz

[ Sign of the Times ] - Touchable holograms

Science Fiction becomes reality: touchable holograms! - An entirely mathematics driven reality, of course. - Invented in Japan.

Importing citations into Mediawiki

I like the format of articles in W'pedia so I decided to use the W'pedia format as my standard. When I add an article about a topic or just a theorem / proof it is important to record a reference to a book, journal article or other source. W'pedia has built quite a system for that which is NOT part of a fresh MediaWiki installation.

When you add a snippet like this in your article in W'pedia all formatting for the citation is automatically done.

{{Citation | title=Name of Book }

It is also possible to create a citation template for a book that is often cited like this

{{BK1 Einstein}}

If you want this functionality in your WikiMedia setup then you have to do three things.
* First you have to Export an example page anywhere in W'pedia using 'Export pages' Make sure that you select to copy templates as well. This page creates an XML file.
* Second, you have to import the XML file into your installation using the Import page. You can find this on Special Pages under Page tools.
* Finally you have to install the ParserFunctions extension.

Preview of the OU Math wiki.

How to set up MediaWiki with MathJax on Xampp

There are two mainstream PHP/MySql servers available: XAMPP and WAMP, a third option which I have worked with quite a lot is The Uniform Server. Before you decide on any software package, always check the user forums and how easy / difficult it is to get help when you have run into trouble. Where XAMPP runs on Windows, Mac, Linux and Solaris, WAMP and Uniform Server are typical Windows solutions.

This time I selected XAMPP and I used the following checklist to install it How-To: MediaWiki on XAMPP, although you can also use this detailed installation guide.

As a last step you have to add MathJax to MediaWiki. You should do this in two steps. First install MathJax and test if it works in your environment by simpling trying to render some of the pages that come with distribution.

Finally, you have to install the MathJax extension for MediaWiki. This step consists of creating and saving a php file and modifying a settings file. Now MathJax should work in MediaWiki. It did in my case as you can see below.

At this stage you can edit your wiki just on your own computer. In a later stage, you might think of sharing your work with other internet users. You can do this by by exporting your mediawiki database and import it to a hosted mediawiki solution. If you like administrating computers and databases you can build MediaWiki again on some host computer, which are usually equipped with Linux. A step in between is that you give internet users access to your site via Opera Unite WebProxy. I am considering this option myself.

Setting up is something you have to do only once. The administration of a wiki site is minimal, i.e. you have to maintain a backup cycle and be prepared to restore a database.

Most of the work in the creation of a Wiki has to do with adding and maintaining content. To use as much experiences of people who did it before you, reading a book on the subject might help. I have read MediaWiki Administrators Tutorial Guide by Mizanur Rahman, and the Missing Manual:



You will see that there are various other books once you start browsing Amazon.

Leonhard Euler's kitchen clock

It is not quite unimaginable that Euler might have had a clock, in his kitchen ( where else? ) resembling the one below. I mean, why shouldn't he?

You might like to solve the crypto-clock puzzle.

[Sign of the Times] - Visitors from abroad

Considering the recent events in the Middle East and by looking at two recent visitors I thought:

Mathematics is universal, not cultural.

Various

Did you like the three-pendulum harmonograph videos? If you did then you will like the website below where you will find mathematical models of it built in excel and very well documented.

I came across a two-word mathematical joke that went like this:


Swine! - Coswine!

If you are into statistics and know how to program javascript then this is probably a must-see website. Add perfect ( statistical ) graphics to a website in a few lines of javascript.

Another hard debunk of the Golden Ratio, the ratio between a Fibonacci number and its predecessor. The link to the work of Leonardo da Vinci was already debunked but it has now been established that there is no link between the Fibonacci numbers and the Nautilus. Read more here.

Video lectures on multivariable calculus ( and more ) ...

I read a commentary on the future of universities.

Education and knowledge have become more accessible in the internet age. Students can do a lot of their work online or at home. But does that mean that universities will become obsolete? Of course not. - Books have to be written, exams have to be prepared and marked, research has to be done and published and students will always want some form of live interaction with a professor or tutor. Even if all lectures would be given in the form of video lectures then these lectures have to produced and as is the case with textbooks, redone every few years. Things have changed and will continue to change. Let's hope for the better.

Some video lectures, I recommended:
- [ NEW ] Multivariable calculus ( Berkeley )
- Video lectures number theory
- [News] - Video Lectures Algebraic Topology ( for Undergraduates )
- video lectures complex analysis

No set of video lectures is better than a well written text-book with lots of examples and exercises. Like this magnificent book on Complex Analysis for example ( I'll start M337 october next year ):

A mathematical clock aka the crypto-clock

I am a great fan of alternative clocks although the clock I like most is the one as can be found at train stations across The Netherlands. To the point.

I have deciphered 9 of the 12 representations so far. And you...?
Update 11/2; deciphered 10 out of 12 of the "crypto-clock".

#50 in the Top 50 best Mathematics Blogs

I made it to #50 in Top 50 best Mathematics Blogs.

Guide To Online Schools.com

Top Math Blogs

I may be last in the list but what counts to me is that this blog is noticed, considered fun and taken seriously.

M381 TMA01 progress

I completed another question of M381 TMA01 Number Theory. Question 4 contained three independent sub-questions about greatest common divisors. The most difficult set of questions I had to make for an Open University TMA to date. ( I am expecting massive scatters of RED on the TMA when it is returned. )

Changes at the Open University

Notice that quite a few courses approach their end of life:
- M337 Final - Oct 2012
- MT365 Final - Feb 2012
- M338 Final - Feb 2012
- MS324 Final - Feb 2012.

This will not affect my plans for 2012 which include MST209 and M336 but I don't think it is useful to plan beyond 2012 with so much changes in the program.

Fibonacci series modulo m

For a while I thought I had discovered some new mathematics. Alas no. Every Coin Has Two Sides. This paper from 1960 by D.D. Wall has no secrets for me. I was well on my way re-discovering and proving until I searched for similar results. I used the Open University library services of course and within a few minutes I found a relevant paper. All this activity is due to NT book 2 from M381.

A Fibonacci series modulo m is cyclic. For example the series mod 3, starting with 0,1 is :
0 1 1 2 0 2 2 1 - 0 1 etc.
and has length 8.

More in the paper.

P.S.
Dream on. I wished that I could travel back in time. To the year 1914 for example. I would go to Cambridge, England. I would take Number Theory lectures from G.H. Hardy and would make friends with Srinivasa Ramanujan. On the way back to 2011 I would stop the clock somewhere in the midst of World War II, to meet Alan Turing to watch him cracking the Enigma code.

What is this: Physics or Mathematics?

A 3-pendulum rotary harmonograph. A visual demonstration that physics = mathematics.



Another video.



What a cool device. I had never seen one before.

M336, M338, M381 merge

Promoted comment from Chris:

Hi Nilo this isn't strictly relevant but I thought you and all
your followers might be interested in the following letter on the
maths course choice forum about the
proposed replacement for the 3 third level courses
M336 Group theory
M338 Topology
M381 Number theory

The idea is to merge it into a 60 point course. I have mixed feelings
but I should be able to complete M338 and M336 next year so it doesn't really affect me. It would seem that the geometric aspects of group theory and topology are being dropped also continuity is going to be explained in sequences rather than open sets which is the mainstay of most of the books on topology and functional analysis I have on my book shelf. On the other hand the fact that a module on rings and fields is going to be introduced is good news. In an ideal world they would keep the current 3 courses with maybe some slight updating and introduce the module on rings and fields as a separate unit. Interesting they don't mention Galois theory so may be they just take you half way up the hill and leave the rest up to you.

Continued here.

Chris mentioned this earlier. My first reaction is that I am glad that I am on M381 in current form. I planned M336 for next year I hope that that does not get mixed up. - ( You need a Plan B. )

[Sign of the Times] - James Stewart

Ever heard of James Stewart? Well his name is on 80% of all Calculus books sold in North America. And when that happens, the money comes pouring in.

Eighty percent market share?! I thought the US was the land of free-enterprise, capitalism, competition. What about all those free, open-source calculus books: they can save the schools a lot of money! I think that somewhere high up in the tree someone is pushing one particular book through everyone's throat.

James Stewart

One of Fermat's other theorems

A Pythagorian Triple (PT) is a list of three numbers $(a,b,c)$ such that $a^2+b^2 = c^2$. Fermat then asked for a PT with the additional property that $a + b$ is also square.

Find a quadruple $(a,b,c,d) $  such that $(a^2+b^2=c^2, a+b=d^2)$
 
( 
Answer:
$a=4565486027761;$
$b=1061652293520;$
$c=4687298610289;$
$d=2372159.$
) 

M381 TMA01 progres

Completed another question today. "Prove that the last digit of a triangular number cannot be 2,4, 7 or 9".

What is your number ?

I would like to share a bit of M381 Mathematical Logic Unit-2 with you.

Every computer program can be emulated on a URM machine which has only four basic instructions: Z(n), S(n), C(m,n) and J(m,n,q). We can give these instructions a unique code as follows:

$Z(n) = 6n-3$
$S(n) = 6n$
$C(m,n) = 2^m3^n+1$
$J(m,n,q) = 2^m3^n5^q+2$

By using this scheme each URM instruction is assigned a unique code. By using these codes and the set of prime numbers it is possible to assign a unique number to every (different) URM program. I.e. to establish a bijection between the set of possible URM programs and $\mathbf{N}$

Conclusion: we can reduce any program to a single integer, and each integer represents a program! ( This conclusion is essential in understanding Goedel's Theorems )

( Do I have readers who still haven't seen Pi, the movie, directed by Darren "Black Swan" Aronofsky? In Pi Maximillian Cohen said, "numbers, everything can be represented by numbers". If every song, every movie, every picture can be mapped to a single integer so can a sequence of pictures. One of the premises in Scientology is that we store at every time unit a dump of sensory images to our memory banks ( vision, smell, sound, touch, taste and a large sequence of emotions ) which form  collectively a detailed recording of our life. All our actions, visions and feelings can be represented by a unique integer as well. You know where I am getting at.  )

We are numbers.

Started on M381 TMA01

This TMA has nine questions. Questions 1-5 on Number Theory and questions 6-9 are on Mathematical Logic.

Completed questions 1 and 2 today up to LaTeX formatting.

Question 1 (12%) is about solving a linear diophantine equation.

Question 2 (10%) is a proof where mathematical induction must be used.

I suppose that any M208 student ( probably a good MS221-er as well ) should be able to do these questions.

The new stuff begins with question 3. For the next session.