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Month: December, 2018

Math Olympiads #1

I’ve been obsessed with Math Olympiads and other kinds of competitive math for a loong time now. Although I should be doing more serious mathematics now that can actually get me a job, I shall ignore good sense and start my series of posts on Math Olympiad problems!

The first problem that I want to write about is a relatively easy one from IMO (International Math Olympiad) 1974. I think it was a long-listed problem, and didn’t actually make it to the test. It asks to prove the fact that

\displaystyle 343|2^{147}-1

(that 343 divides 2^{147}-1)

The important thing in this problem is to figure out that 343=7^3 and 147=3.7^2. Hence, we’re being asked to prove that 7^3|2^{3.7^2}-1.

We have the following theorem for reference: If a and b are co-prime, then a^{\phi(b)}\equiv 1\pmod b, where \phi(b) is the number of natural numbers that are co-prime with b. Fermat’s Little Theorem is a special case of this.

Now on to the calculations and number crunching. We have fleshed out the main idea that we’ll use to attack the problem. We can see that \phi(7^3)=7^3-7^3/7=6.7^2. Hence, as 2 is co-prime with 7^3, 2^{6.7^2}-1\equiv 0\pmod {7^3}. We can factorize this as (2^{3.7^2}+1)(2^{3.7^2}-1)\equiv 0\pmod {7^3}. Now we have to prove that 7^3 does not have any factors in common with 2^{3.7^2}+1. As 2^3\equiv 1\pmod 7, 2^{3.7^2}\equiv 1\pmod 7 too. Hence, 2^{3.7^2}+1\equiv 2\pmod 7. This shows that 2^{3.7^2}+1 does not have any factors in common with 7^3, and leads us to conclude that 7^3|(2^{3.7^2}-1), or 343|2^{147}-1. Hence proved.

I realize that the formatting is pretty spotty. I shall try and re-edit this in the near future to make it more readable.

 

The Möbius strip

Almost every Math student (or even otherwise) has heard of the Möbius strip at least once in their career. I too have. Embarrassingly, I always had my doubts about it. And I can bet that a lot of people have the same problem.

A Möbius strip is almost always taught in the following “intuitive way”: take a long, thin strip of paper, and connect the opposite ends along the length in such a way that one end is oriented in the opposite direction as compared to the other end. You get something that looks like the picture below:

Paper mobius strip

And when you take an object, say a pen cap around the Möbius strip, you’re somehow supposed to believe that the pen cap returns to the same spot as before, but with the opposite orientation. But you (or at least I) would think “No!! The pen cap has not returned to the original starting point! It is at the other side of the piece of paper! Points at opposite sides of the paper can’t be the same point!”

As I grew older, I decided to exorcise my demons, and think about these seemingly fundamental concepts that I had accepted, but not really understood. And I have arrived upon the conclusion that the paper model is completely wrong, at least to convey the essence of the Möbius strip. A much much more convincing model is the following:

Mobius strip

Imagine a coin that is traveling from top to bottom. The coin is colored in the following way: the left side is red, and the right side is blue. After disappearing at the bottom, it re-emerges at the top, but now its left side is blue, and its right side is red. This picture is much much clearer to me than paper strips forming loops.

Thus ends my spiel for the day!

Term paper on Conformal Geometry

I had to write a term paper on Conformal Geometry for my Differentiable Manifolds class. Although I slacked off on it at first, I’ve spent the whole day today trying to pretty it up. Although I couldn’t make some changes like changing the font of “Abstract”, or inserting a picture on the cover page, mainly because I didn’t understand some aspects of the layout I had downloaded from the internet, I’m still reasonably proud of how the paper looks (the content could have been better for sure). I’m uploading it for reference.

Here’s the paper-Introduction to Conformal Geometry