Fermats Last Theorem

30.05.10

Fermats last theorem is a problem in number theory stating that the equation
$$a^n + b^n = c^n$$
can not be satisfied by three positive integers, a, b and c for n > 2. For n = 1 and n = 2 we have infinitely many solutions, for instance

(n=1): 1 + 2 = 3

(n=2): 3^2 + 4^2 = 5^2

The theorem was conjectured by the mathematician and lawyer Pierre de Fermat in 1637 and wasn't successfully proven generally (for all n) before 1995(!) over 350 years later. The conjecture was found written in the margin of his own copy of Arithmetica (written by Diophantus) stated in the following way:


I have discovered a truly marvelous proof that it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. This margin is too narrow to contain it.


Fermat did however fail to write down his so called "truly marvelous proof". He only proved the special case for n = 4, a case which was already proven by Leonardo Fibonacci in 1225. This started the spectacular quest for a rigorous proof which motivated the development of both algebraic number theory and the proof of the modularity theorem. Mathematicians were first able to reduce the problem to only checking for exponents which were prime numbers. The progress was slow at first, by 1838 the conjecture was only proven for the primes 3,5 and 7. Things really started to kick of when Ernst Kummer proved the conjecture for a (possible infinite) class of primes known as the regular primes (1850). This result allowed mathematicians to prove the conjecture through computer studies for all odd primes up to four million.

The next piece of the puzzle was came from a Japanese mathematician by the name Yutaka Taniyama who is most famous for the Taniyama-Shimura conjecture. He was unfortunately a man who suffered from depression and he tragically commited suicide in 1958 before knowing the real significance of his conjecture. It was the mathematician Kenneth Alan Ribet who in 1986 (28 years after Taniyamas suicide) proved that the solution of the Taniyama-Shimura conjecture would imply that Fermat's conjecture was correct. This result inspired the mathematician Andrew Wiles who was finally able to prove the conjecture for all n in 1995, putting the perhaps most famous conjecture in the history of mathematics to rest.

If you would like to read more about this fascinating story, you should check out this book.


Pierre de Fermat (1601 - 1665)

Awesome Algebra!

28.05.10

Today I had my final exam in algebra and number theory, and it got me thinking. A significant part of the curriculum of that course came from research done by Norwegian mathematicians. I therefore present three important contributors to the exiting world of algebra. You can click on their names below their pictures to open a Wikipedia article for more in-depth reading.


Peter Ludwig Mejdell Sylow


Sophus Lie


Niels Henrik Abel

Grahams Number

27.05.10

Following yesterdays entry about really large, finite numbers, I hereby bring you the exiting follow-up. But before we can talk about larger numbers, we need some new tools to describe them. We will borrow those tools from Donald Knuth who introduced Knuth's up-arrow notation in 1976. This entry will be loosely based on the Wikipedia article found here.

Firstly, multiplication can be defined as iterated addition as follows:

a * b = a + a + a + ... + a

You add a together b times.

Next, you can define the power of a number as iterated multiplication as follows:

a^b = a * a * a * ... * a = a↑b

You multiply a together b times. The last equivalence shows how powers are represented in Knuth's up-arrow notation.

We've now seen how a single arrow in Knuth's up-arrow notation works, so let's extend it to two arrows. What does the following mean?

a↑↑b

This means that we start to nest powers as such:

a↑↑b = a^a^a^...^a

You take the a'th power b times.

This is indeed a quickly growing function. Some numerical examples follow below:

2↑↑2 = 2^2 = 4
2↑↑3 = 2^2^2 = 16
2↑↑4 = 2^2^2^2 = 65 536
2↑↑5 = 4 294 967 296
2↑↑6 = 1.84 * 10^19

But that's just two arrows! Let's add another one and see what happens:

3↑↑↑2 = 3↑↑3 = 3^3^3 = 7 625 597 484 987
3↑↑↑3 = 3↑↑3↑↑3 = 3↑3↑(7625597484984 times)↑3

Oh dear!

As you might have realised by now, Knuth's up-arrow notation is able to cope with enormous numbers in a way normal notation can't. Keep this in mind when we move on to the big fish.

Grahams Number

Grahams number is large. Really large. So large that Knuth's up-arrow notation starts to have problems displaying it. In that notation it can be written as



where the number or arrows in the n'th layer is determined by the value of layer n-1. I have huge problems wrapping my head around the magnitude of this number, but it helps to start with easy examples of Knuth's up-arrow notation and slowly building your way up from there. This number is an upper bound on the solution to a problem in Ramsey theory, which is a topic that I am not familiar with at all and therefore do not dare to write about. But one thing is for sure, that is an insanely large number!

I don't know what else to say. It is just... enormous... gigantic... the English language lack proper words to describe it.

Well. If you would like to read more about Grahams number and Knuth's up-arrow notation, check out the following links
Graham's number
Knuth's up-arrow notation

Now I need a cup of coffee to calm my head.

Large Numbers (!!!)

26.05.10

As a (soon to be) mathematician, numbers are of great interest to me. This idea of a number as something you can attach to for example, a group of objects in real life, has proven to be very powerful in everyday life as well as all branches of science.

But how large can a number get? How large numbers can our brains comprehend? The funny thing is that there isn't really a "largest" number. You can think of the biggest number you can and always get something bigger by adding one. And then adding one to that number. And so on and so on, which leads us to the concept of infinity. Infinity can, in a way, be easier to grasp than really large numbers. Infinity can be thought of as the "number" you get if you keep adding one to a number for eternity. In our finite universe, however, you will fall short. You can't assign infinity to a group of objects in everyday life the same way as you can with finite numbers. For instance, twelve apples make sense while an infinite amount of apples makes your head spin.

From what I've said so far you may get the notion that finite numbers are reasonably nice to handle. The spinning of the head caused by the concept infinity is strangled - you can count one plus one plus one for a finite amount of time and reach the finite number.

Now, how large can a finite number get before our brain starts to spin again? I will try to visualise powers of ten to illustrate.

1 (10^0)
One. Starting off easy. Just think of something like a car or a house, and you've visualised the number one.

10 (10^1)
The number of fingers you (should) have.

100 (10^2)
The number of fingers of you and nine of your friends.

1000 (10^3)
The number of pages in a fairly large book.

10.000 (10^4)
Approximately the number of seconds in three hours.

100.000 (10^5)
The number of hairs on an average human head.

So far we're doing alright. Let's crank up the power dial. What about ten to the power of 25? One, with twenty five zeros after it.

10.000.000.000.000.000.000.000.000

This number is approximately equal to the number of H2O-molecules in one liter of water. Simply stunning. But let's not stop there! What about ten to the power of 100? One, with a hundred zeros after it. This number is refered to as a Googol and written out looks like this:

10.000.000.000.000.000.000.000.000.000.000.000.000.
000.000.000.000.000.000.000.000.000.000.000.000.000.
000.000.000.000.000.000.000.000

Is it equal to the number of people on earth? No.
What if you took the shoes of all the people on earth? No, still not enough.
What if every person on earth took a liter of water and we added together all the H2O-molecules?
What about all the stars in the observable universe?

No, no, no. We still fall short. We have to turn the dial back again to about ten to the power of 80. We then reach an estimate for the number of atoms in the observable universe. Indeed an insane amount to count.

Well, we've come this far, so let's push on. What if we take ten to the power of a Googol? Or, ten to the power of ten, to the power of a hundred. This number is called a Googolplex. A ginormous number that you have no chance of spelling out the zeros of. There simply isn't enough space in the universe. That, and you would not have enough time. If you were able to write out two digits per second, it would take you about ten to the power of 80 times the age of the universe.



Phew, now these are large, finite numbers which really makes your head spin. But a Googolplex is far from the biggest number out there. Far from it. Next time I will investigate a number called Grahams number - a number so large it makes Googolplex shrink into the corner by comparison. Exciting indeed!

But until then, watch this video on YouTube by Carl Sagan, explaining the vast difference between a Googol and a Googolplex.

Solution to the Fuel Problem

21.05.10

As a follow up to this entry, I present this response, posted on the same forum.

The post was as follows:

Alright, I keep seeing bullshit around here with magnets and cars. They're always implemented in the wrong way and you're left with prototypes for cars that wont even move an inch. Seriously it's like these people coming up with these dropped out of middle school or something

This is my plan

The axle would be connected to the wheels, just to clarify.

Let's solve the world fuel problem.



Great fun! :D

Take it away, Schrödinger!

19.05.10

As a lot of my entries here on this site has been about the Schrödinger equation (see this, this, this and this), I now, as the project has been graded, feel that I can release it for you to read. It really was a lot of work to put this together and I am very pleased with the result we (me, Bjørn Theisen and Erik Bakken) were able to produce.

Attached to this entry you can find the report, the poster we used when we presented our work (in Norwegian), and the MATLAB-code used to actually solve the equation. I do not know if the code runs well under OCTAVE but I would at least give it a go if you're interested in seeing some awesome plots and don't have MATLAB installed.

Downloads:
- The report
- The poster
- The code

... now all that is left is the exam. :D

Take it away, Schrödinger!

Congratulations, Norway

17.05.10

Today it is May Seventeenth, the Norwegian constitution day.



So leave those books at the study hall today and have fun celebrating!

Bacon

15.05.10

The Internet, as well as almost everyone else on the planet, loves bacon. This is why you can find recipes for almost anything combined with bacon online. It's so easy! Just take your favorite meal and type 'name of meal + bacon' into Google and off you go to another greasy adventure.

NOW

Did you know that household appliances and road signs also love bacon? Below are some pictures taken around the world showing inanimate objects, such as automatic dryers and road signs, giving sound bacon advice.


A public bacon dispenser.


Another public bacon dispenser.


Caution: Do not fuse yourself with bacon.


This picture speaks for itself.


Do not submerge your car into bacon!

Dear Desk

14.05.10

As a student, my desk at the university is a place I value greatly. Our institute actually assigns each student their own desk at the start of the third year. A gesture which is highly appreciated. At the start of the semester, the desks start out as bland as bland can be, but during the course of the semester they tend to get more personalized. There are however certain things which the desks share in common. Below is a list describing some of those similarities so that you can recognize when you're getting close to the desk of a student studying mathematics.

You should at least find:
- A ridiculous amount of instant coffee
- Stacks upon stacks of paper
- Loads of thick, mathematical books
- At least one calculator

You can also find:
- Food which only require hot water to prepare
- Strange web comics you've never heard about or seen
- At least one marker pen
- A mini blackboard
- Coffee mugs or other kitchen appliances containing mathematical formulas

Below you can find two pictures of my own desk. See if you can find any of the things mentioned on the list. You can click on the pictures to see a larger version.

So how is your desk arranged? Do you feel familiar with my desk or have you gone in an entirely different direction?

Brilliant Blackboards

13.05.10

Between the computer labs and study halls here at the university, we have a blackboard at our disposal. This can be both useful and great fun. Below are some pictures taken during the last couple of weeks. As you can see the exam revision period is beginning to take its toll on people. :D

Slependen?

12.05.10

After inspecting some traffic statistics for my site, I came across an interesting piece of information. Most of my visitors come from Trondheim (which is natural), but following in second place is a place called Slependen.



Since I didn't know much about Slependen, I decided to give Google the job of educating me. According to Wikipedia, it houses lucrative stores such as IKEA and Smart Club. And that was about it

I don't think I know any people from there, so that it should generate that much traffic is indeed peculiar.

So, people of Slependen, make yourself known! Who are you?

Braid

11.05.10

As the exam revision period is charging me at full force, I did what any other sensible student would do, I purchased a new game from Steam to keep myself occupied. That'll teach me to be.. too.. tangled up in school.

:D

The title of the game is Braid, short and simple, and I can not begin to describe how much fun it was to play. If you are into puzzle games at all, you are sure to love it as much as I did. In short, it is a platform puzzle game where your only objective is to collect pieces for a jigsaw puzzle, which sounds simple enough, right? But you haven't heard the best part yet!

YOU CAN CONTROL TIME!!

That's right. You didn't misread. If you die, you can scroll back time to avoid whatever it was that killed you. And I know what you're thinking now, "oh my, that sounds like CHEATING", but you couldn't be further from the truth. To preform well in this game you are required to master the skill of time manipulation, that is, bend time and events to fulfill your greatest goals (which, oddly enough, happens to be collecting jigsaw pieces). Each level brings a new time-bending element into play. For instance, in one level, time moves in the direction you're moving. If you're walking right, time moves forward and if you're walking left, time moves backwards. Spooky! Another level allows you to drop an orb which locally slows down time. Now how about that.



So what are you waiting for? Go fetch it! :D

Spilling Coffee for Science

10.05.10

As mentioned in this post, I've conducted an experiment investigating the spilling of coffee in everyday life. As the project has been completed and graded, I can now publish it here without being charged with plagiarism by myself.

So without further ado, here is the project. Hope you like it.

Physics in Movies

08.05.10

While surfing mindlessly about the net today, I came by the following picture:



The discussion where the picture was posted, was revolving around whether Goofy would move or not. Needless to say, if he did move, we would all be enjoying free energy by now. Since Goofy is holding on to the magnet via the rod, there is no external force acting on the system and hence he stays in place (in accordance to the first law of Newton).

After reading further on in the discussion I was not only amused by all the mind blowing answers (read: people who claimed to have defied the laws of Newton with their home-built zero-net-force-mobile-car or whatever they called it), it also got me thinking about why people were so quick to embrace this idea as plausible.

And, to be as cliche as possible, I blame Hollywood! :D
9 Laws of Physics That Don't Apply in Hollywood

Very entertaining reading indeed, so check it out.

EDIT: Found this picture as well. Awesome idea indeed. :D

Final Schrödinger

07.05.10

As the term project in TMA4212 is done and the poster presentation was a success, I feel that the non-linear Schrödinger equation can make it's way to the back of my brain for some well-deserved rest.

There is, however, a final animation which we used for the presentation that I want to share with you. It shows the solution with initial condition

u(x,0) = sqrt(2) * sech(x)

starting on the time interval [0,1], incrementing 1 each frame and ending on [0,150]. The axis stay constant as I am using 500 nodes in space and time for each discretization (rendering the final approximation more inaccurate than the first, but hey, for this demonstration it is sufficient).

The solution is approximated using a Crank-Nicolson finite difference scheme implemented in MATLAB.

To watch the animation, click the image below:

(2.8MB)

If you liked that, you might also want to read this.

So Where Am I Heading Anyway?

06.05.10

After spending almost three years at NTNU, I think I finally know what area I want to specialize in. Numerical mathematics. For those of you who don't know much about this field, I will give a small introduction to get you up to speed.

As a numerical mathematician I am more interested in finding approximations to problems rather than solving them exact. The approximation can be of great interest if, for instance, the exact solution doesn't exist and I have to consider, what is the closest I can get to a solution.

Other situations where a numerical approach is more applicable, is when you have an enormous mathematical system, like for instance the model simulating the day-to-day weather in Norway. It would simply be too much work (if even possible) to solve such a system exactly. We do not possess adequate tools in functional analysis to efficiently cope with systems of partial differential equations of that magnitude. A numerical approximation algorithm can, however, be installed in a supercomputer (which is the case here in Trondheim) and it can approximate the system to a user-defined accuracy.

A good numerical algorithm allows the user to get as close as possible to the wanted solution. In other words, if you crank up the accuracy, the error should decline systematically. How fast this error declines defines how "good" your method is. Preferably your method should allow for machine accuracy within a reasonable runtime.

This field is, as I hope I've indicated, a very applied field. Below is a small list (from Wikipedia) where numerical mathematics is used.

• Advanced numerical methods are essential in making numerical weather prediction feasible.
  
• Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations.
  
• Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically.
  
• Hedge funds (private investment funds) use tools from all fields of numerical analysis to calculate the value of stocks and derivatives more precisely than other market participants.
  
• Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs.
  
• Insurance companies use numerical programs for actuarial analysis.
  

But before any of that can happen I have to complete the exams for this semester. Which means I should probably start studying about now.


The supercomputer in Trondheim, used to calculate weather predictions.

So where are you going? What courses / jobs will you be doing in the autumn of 2010?

SUPER MEGA COMICS

05.05.10

Everyone should read this excellent web comic I quite recently came over.

http://www.supermegacomics.com/

I should warn you though, don't try to read too much into each strip. :D

What's this, I don't even-

03.05.10

Snow? SNOW?



That's right. Snow. Snow in MAY. I woke up today just to find that winter wasn't part of the past just yet. Oh no, when you live in Trondheim, the weather plays games with you. You can be caught walking about, minding your own business in the sun, when oh my god suddenly a wild snowstorm appears out of nowhere.



But what can you do. I tried yelling and punching at the snow but it was all in vain. I guess you are stuck with the random weather when you choose to live in this city.



... but it isn't all bad! Maybe tomorrow I can craft a snowman before I go to campus. :D