Pythagorian triples were known in 1900BC

Just read In A primer of analytic number theory by Jeffrey Stopple, 2003 that there is a Babylonian cuneiform tablet (designated Plimpton 322 in the archives of Columbia University) from the nineteenth century b.c. that lists fifteen very large Pythagorean triples; for example, 127092^2 + 135002^2 = 185412^2.

This means that they must have known how to generate these numbers with
x = 2*s*t, y = s^2 − t^2, z = s^2 + t^2

For s=2,.t=1 we get the well known x=4, y=3, z=5 and
for s=3, t=1 we get 6, 8, 10 and
for s=4, t=2 we get 16, 12, 20.

1900 BC.

So 1900BC is much closer than I thought. The current blazing speed of scientic development has only been reacned since the last few hundred years or so.

( Test Blogger Buddy )

Test. This is a post from within the Vista Sidebar using Blogger Buddy.

Blog or diary?

I think the words 'blog' and 'diary' contradict each other. This should be either a blog or a diary now it is an attempt to be both while the endresult is neither. My blog/diary contains a lot of introvert babble like 'TMA done'. Who cares? So I completed MST121 TMA02 early this year. The course is completed. Done. I am going to write more about mathematics.

With the MS221 in less than two weeks I'll be free to study whatever I like until at least february 2010 when I will be doing 'Pure Mathematics' and 'Groups and Geometry'. I think I am going to self-study Introduction to Analytic Number Theory by Apostol. That will prepare me for three OU courses: Number Theory (M381) and Analytic Number Theory I ( M823 ) and II ( M829 ).

I would very much like to be able to prove, or at least understand the proof of, The Prime Number Theorem.

Thinking of the future

Received the Open University Prospectus Mathematics and Statistics 2009 / 2010 today. Although still in the beginning of my studies I already more or less planned my undergraduate studies so I looked at the MSc. programme. It is possible to study for a M.Sc. degree after you have finished the 360 points B.Sc. course. The M.Sc. course requires 180 points from five 30 point modules and a 30 point dissertation. With my interests I probably do Analytic number theory I and II ( 2 * 30 ), Coding Theory ( 30 ) so still 60 to choose and the Dissertation ( 30 ) of course.

We'll see.

Ramanujan's magic square formula

Today I started to read the Ramanujan biography ( The e-book version, of course. ) The book looks promising. What was it like to communicate with someone gifted with such powerful mathematical insights? I am hoping ( a bit less after every failed try ) for a book that pulls me in that world of it's own. I have been there before on many occassions, while reading other books of course. Scientologists have a special word for it: exteriorizing.

The Man Who Knew Infinity.

At a very young age he designed the following formula for a 3 by 3 magic square:

C+Q | A+P | B+R
A+R | B+Q | C+P
B+P | C+R | A+Q

where A,B,C are integers in arithmetic progression and so are P,Q,R.

Try it, it works!

MS221 - TMA04: Done

I reviewed MS221 - TMA04, made some minor corrections and decided that the A is ready for shipment to T. I nevertheless wait with shipping the TMA ( Cut-off date is 30 sep anyway ) because I haven't received TMA03 back yet. If I get a high enough mark on that one I might even get away with a low score ion TMA04. We'll see.

MS221 exam close by

I copied this from a post by "Tma Machine" :

Here’s how a rough outline of what’s in the 2008 paper (hopefully this won’t constitute copyright infringement!):

Part 1
Question 1: Finding a closed form for a recurrence system
Question 2: Identifying and sketching a conic
Question 3: Stating the rule for some isometries, and for a composite isometry, and then using the double-angle and half-angle formulas to show a result
Question 4: Classifying fixed points of a curve, then sketching the graph of the curve and using graphic iteration construction
Question 5: Identifying basic linear transformations, applying them to a vector, and stating an invariant line for each one
Question 6: Finding eigenvalues, eigenlines and eigenvectors.
Question 7: Differentiation
Question 8: Integration
Question 9: Finding and manipulating Taylor series about 0, and classifying stationary points
Question 10: Finding the modulus and argument of a complex number, converting them from Cartesian to polar form and vice versa, and using the formula for powers of complex numbers.
Question 11: Using Euclid’s Algorithm and working with exponential ciphers.
Question 12: Combining variable propositions, finding a case for which a given proposition is false, and finding the converse of a proposition.

Part 2
Question 13: Looks like it’s about conics, but I haven’t done this one yet.
Question 14: Linear transformations
Question 15: I haven’t done this one yet either, but it looks like it involves differentiation, integration and stationary points.
Question 16: Groups

I sort of expected this, so I am not surprised. But I am not entirely ready for the exam yet. I still have several weaknesses but I am confident I can handle these in time. To be honest I wished the exam was over and done with. It is obviously my objective to score in the 90-100 because that is what I score for TMAs. But I check, doublecheck, re-check my TMAs at least two times. That makes a lot of difference.

Logic puzzle

In lecture 2 Dr Kamala describes a murder mystery which can be solved entirely using logic in a Sherlock Holmes or Hercule Poirot like manner! Very nice.

Discrete Mathematical Structures

Lectures on YouTube by Prof. Kamala Krithivasan, Department of Computer Science and Engineering, IIT Madras.

Beautiful lectures for the whole planet to watch.

Excercise

I think this is a nice exercise at the MST121 / MS221 level.

Show that : Sin ( 18 degrees ) = 1/4 * ( SquareRoot(5) - 1 ).

Will use it to practice the trig formulas for the MS221 exam.

Sage or Mathematica ?

Sage and Mathematica are both very powerful mathematics tools. For an absolute beginner both packages will do, I suppose.

Mathematica is closed source and for the price of a full edition Mathematica package you can also buy a state-of-the-art Sony VAIO or other sexy netbook or tablet you are thinking of buying. The Mathematica programming language is a so called propriety language. That means that it is only used in Mathematica and the owners of Mathematica can change the language whenever they want. Mathematica has a steep learning curve. Now that I master the basics of it I am very satisfied with what Mathematica can do for me.

Because Sage says it is possible to integrate Mathematica into Sage ( haven't figured yet out how, I suppose I need the Mathematica for Linux version ) I am having another look at Sage. Sage is Linux only but you can run it entirely from within a browser. So you can install Sage on a Linux server ( for example in a VMWare virtual machine ) and run it on Windows in your browser.

( More later about my Sage findings. )

http://www.sagemath.org/

Characteristic of a ring

The characteristic of a ring is the number of times you must add the multiplicative identity element in order to get the additive identity element. If adding the multiplicative identity element to itself, no matter how many times, never gives us the additive identity element, we say the characteristic is 0.