A simple proof of Pythagoras Theorem

Narasimhanna sent this simple proof of Pythagoras theorem. I have drawn pictures that make it easy to understand the already simple proof. If you have questions post them in the comments section.

A lecture on prime numbers

Here is a very accessible lecture for a general audience, on prime numbers, by a very bright young mathematician.
   http://www.youtube.com/watch?v=PtsrAw1LR3E

In the beginning of the video observe the description of his profile.

The key result of his that he talks about is the following:
"Primes contain arbitrarily long arithmetic progressions"

Examples:
   3,5,7 are primes. These are an arithmetic progression of 3 prime numbers spaced 2 apart.
5,11,17,23 are an arithmetic progression of 4 prime numbers with common difference 6.
7,157,307,457,607,757 are an arithmetic progression of 7 prime numbers (the common difference is 150).

The result he proved in 2004 says that among primes, there are arbitrarily long arithmetic progressions. The longest found so far contain 23 prime numbers:
56211383760397 + 44546738095*n  ( n = 0, 1, ... , 22)

I am sure most of you will agree that understanding the result is easy. Notice, in the lecture, that he mentions that there can be arithmetic progressions millions of prime numbers long; no one knows how to find them!!  The best we have got is 23 primes sequence.

 Also, visit his blog http://terrytao.wordpress.com/ and check out the "Career Advice" section for something you may find useful.

WHEN EMPERORS BUILD GREAT PALACES, WE CAN HAVE A ROLE TOO!

Srinidhi's question:

Peddappa, what is your subject of research?


Narasimhanna's reply:

Dear Srinidhi,

     This is a more difficult question to answer and perhaps not very important.

 

I will write about it some time later.

 

You see, all this is a small part of a great human enterprise.

 

It is evolution of ideas/understanding  about various things around us that really makes our lives easier, more comfortable and satisfying, not just as individuals but also in a broader context of communities, societies and even civilizations.

 

The various issues about which one may have to think about, for example,

- life itself,

- nature of man, mind, intelligence,

- the way things are, like how the world around us work, our own body for example,

- the way things change,  

- the way we can adopt ourselves, adopt our institutions around us in view of the growing understanding of such matters

- the suitable tools that we can probably develop to make our lives easier (technology!)

- develop suitable social institutions, like schools/universities, legal system, banking system, etc

- and so on

 I am sure you can add to the list.  Anyway, you get the idea.

 

All scientists, philosophers and thinkers in general generally think about these issues.

 

As you very well understand, each of them is too huge for any individual or even a group of people.

 

So even great thinkers address a very small part of at most one of these aspects. 

 

People like us are like small helpers in this great endeavour. You know a great man once said

 that WHEN EMPERORS BUILD GREAT PALACES, WE, THE SWEEPERS ALSO HELP!! 

 

 

The emperors he was talking about were people like: Budha, the Shankaracharya, Newton, Einstein, Darwin, Christ, Mohammed, Moses, Confuscius, Socrates, etc.

 

You know we do not even know the names of many such `EMPERORS'  in very distant, forgotten civilizations (like Mayans, like many societies in Africa lost to history, preroman European civilizations,  etc)

 

But people like us (you and me!!)  are also important in this enterprise because: 

 we keep these activities alive in the society we live in  (mostly as students, as  teachers and transmitters of these ideas to new generations) .

 

This is why teachers are very important in a society and our forefathers gave such an exhalted position to our GURUS.

 

 

Yours affectionately,

-Narasimha