fermat’s last theorem: 1

•September 27, 2009 • Leave a Comment

Fermat’s Last Theorem is probably one of the most famous problems in number theory. The statement itself looks simple – someone who has some knowledge about exponents and addition should be able to understand the theorem. The theorem states that

x^n+y^n=z^n

has no trivial (nonzero) integer solutions x, y, z when n > 2 is an integer. Yet, ever since the 17th century, when Pierre de Fermat proposed it, the top mathematicians were neither able to prove it nor give a counter example. That is, until Andrew Wiles was able to prove it in 1995 (his 1993 proof contained an error) using Galois representations. What makes it even more interesting is that Fermat claimed that he had a prove of this theorem. However, it had to be a fairly elementary one, since most of the mathematics that Wiles used was not even present in the 19th century, let alone during the 17th.

In this series of posts, we’ll try to chronicle the works of different people who tried to solve this problem. We’ll even try to prove it for a few certain cases.

The next post will be about Pythagorean triples, which most of us should be familiar with. Pythagorean triples are numbers which satisfy the Pythagorean Theorem:

x^2+y^2=z^2

We’ll try to create derive Euclid’s formula for generating such triples, which will be very useful when we try the formula for other values of n

books

•September 18, 2009 • Leave a Comment

I was going to make a post about my personal goals for this year, but I realized that there are way too many of them!! I want do this, I want to do that…it’s never going to end. I guess I’m just going to talk about the books I’m planning to read this year instead. There are quite a few of them, although a number of them are mathematical literature kind of (nonfiction/biography). My school workload will probably increase threefold or greater as the school year progresses, and that’s my priority over anything at this point, so my time outside of school to do anything extra would be limited a bit. Anyways, here are the books I’m hoping to get through this year:

1. A Beautiful Mind by Sylvia Nasar (and the movie)

A Beautiful Mind - Sylvia Nasar

I’ve read quite a bit about the great John Nash, but I want to know more about his wonderful and brilliant life and mathematical career. I’ve also heard that this is a great book, and also read some positive reviews about the movie.

2. A Mathematician’s Apology by G.H. Hardy

A Mathematician's Apology - G.H. Hardy

Someone recommended this to me, and it seems like a good book, by one of the greatest mathematicians of the recent times. Should be a good read.

3. The Calculus of Friendship by Steven Strogatz

The Calculus of Friendship - Steven Strogatz

I came across this book after browsing randomly on Amazon. I read the review and it seems like a good book. Check out the review of one of the readers:

This is an absolutely beautiful book about the relationship through letters of a high school math teacher and one of his students who becomes a world class mathematician. It’s about their lives and the mathematics that bound them together. I have read quite a few memoirs and don’t recall any that choked me up like this one did, I want to thank Dr. Strogatz for being so open. Also, the math in the book is very interesting and well explained, if I could give if more than five stars I would.

4. Introduction to Electrodynamics by David J. Griffiths

Introduction to Electrodynamics - David J. Griffiths

The classic undergraduate E&M text by the great Griffiths. I haven’t worked on physics beyond the introductory texts of Halliday-Resnick-Krane and the Feynman Lectures, and I’m planning to do soon. This should be a good start.

5. Calculus by Michael Spivak

Calculus - Michael Spivak

The epitome of all Calculus books (along with Apostol’s text and Courant’s text), I’ve finally gotten hold of this gem near the end of this summer and have started working on the exercises. And I must say, this is one interesting and fascinating book!! The exercises really make me think, and I’m only on the first 2 chapters of the book right now. This is probably going to take most of my time outside of school, but hey, it’s worth it!! After reading this book, I’ll probably going to work on a full-blown analysis text, and people say that this book should do more than enough job to prepare me for it.

So yeah, so far those are the books that I want to read and work through throughout the school year. Griffith’s E&M and Spivak’s Calculus should take up most of my time outside of school. I’ll probably read some other minor books (both math and not) in my spare time (if any), and I’ll write about them when I get the chance.

aqueronte

•September 13, 2009 • Leave a Comment

Today was certainly not a good day for me. Witnessed Nole helplessly get butchered by Federer. Hopefully Del Potro gives him a good serving of forehand winners tomorrow during the championship match.

Anyways, I’d like to share with you some really good Spanish literature. It’s called “Aqueronte”, written by Jose Emilio Pacheco. Good stuff, indeed.

Son las cinco de la tarde, la lluvia ha cesado, bajo la húmeda luz el somingo parece momentáneamente vacío. La muchacha entra en el café. La observan dos parejas de edad madura, un padre con cuatro niños pequeños. Atraviesa rápida y tímidamente el salón, toma asiento en el extremo izquierdo.

Por un instante se ve nada más la silueta a contraluz del brillo solar en los ventanales. Se aproxima el mesero, ella pide una limonada, saca un block de taquigrafía, comienza a escribir algo en sus páginas. De un altavoz se desprende música gastada, música de fondo que no ahogue las conversaciones (pero ocurre que no hay conversaciones).

El mesero sirve la limonada, ella da las gracias, echa un poco de azúcar en el vaso alargado y la disuelve haciendo girar la cucharilla de metal. Prueba el regresco agridulce, vuelve a concentrarse en lo que escribe con un bolígrafo de tinta roja. ¿Una carta, un poema, una tarea escolar, un diario, un cuento?. Imposible saberlo, como imposible saber por qué está sola ni tiene a dónde ir en plena tarde de domingo. Podría carecer también de edad: lo mismo catorce que dieciocho o veinte años. Hay algo que la vuelve excepcionalmente atractiva, la armoniosa fragilidad de su cuerpo, el largo pelo castaño, los ojos tenuemente rasgados. O un aire de inocencia y desamparo a la pesadumbre de quien tiene un secreto.

Un joven de su misma edad o ligeramente mayor se sienta en un lugar de la terraza, aislada del salón por un ventanal. Llama al mesero y ordena un café. Luego observa el interior. Su mirada recorre sitios vacíos, grupos silenciosos, hasta fijarse por un instante en la muchacha.

Al sentirse observada alza la vista, la retrae, vuelve a ocuparse en la escritura. Ya casi ha oscurecido. El interior flota en la antepenumbra hasta que encienden la luz hiriente de gas neón. La grisura se disuelve en una claridad diurna ficticia.

Ella levanta nuevamente los ojos. Sus miradas se encuentran. Agita la cucharilla, el azúcar asentado en el fondo se licua en el agua de limón. Él prueba el café demasiado caliente, en seguida se vuelve hacia la muchacha. Sonríe al ver que ella lo mira y luego baja la cabeza. Este mostrarse y ocultarse, este juego que los divierte y exalta se repite con variantes levísimas durante un cuarto de hora, veinte, veintico minutos. Hasta que al fin la mira abiertamente y sonríe una vez más. Ella aún trata de esconderse, disimular el miedo, el deseo o el misterio que impide el natural acercamiento.

El cristal refleja, copia furtivamente sus actos, los duplica sin relieve ni hondura. La lluvia se desata de nuevo, ráfagas de aire llevan el agua a la terraza, humedecen la ropa del muchacho que da muestras de inquietud y ganas de marcharse.

Entonces ella desprende una hoja del block, escribe ansiosamente unas líneas mirando a veces hacía a él. Golpea el vaso con la cuchara. El mesero se acerca, oye lo que dice la muchacha, y retrocede, gesticula, da una contestación indignada, se retira con altivez.

Los gritos del mesero han llamado la atención de todos los persentes. La muchacha enrojece y no sabe cómo ocultarse. El joven contempla paralizado la escena que no pudo imaginar porque el lógico desenlace era otro. Antes que él pueda intervenir, sobreponerse a la timidez que lo agobia cuando se encuentra públicamente a solas sin el apoyo, sin el estímulo, sin la mirada crítica de sus amigos, la muchacha se levanta, deja un billete sobre la mesa y sale del café.

Él la ve salir sin intentar ningún movimiento, reacciona, toca en el ventanal para pedir la cuenta. El mesero que se negó a trasmitir el mensaje va hacia la caja registradora. El joven aguarda angustiosamente dos, tres minutos, recibe la nota, paga, sale al mundo del anochecer en el que oscurece la lluvia. En la esquina donde se bifurcan las calles, mira hacia todas partes bajo el domingo de la honda ciudad que ocultará por siempre a la muchacha.

my goals for the year

•September 11, 2009 • Leave a Comment

I haven’t been posting in the last few days, this week has been hectic (for me, at least). I’m still adjusting to “back to school” attitude, and consequently, workload. In this post, I guess I just want to share my goals for the following school year, scholastic goals in this post and personal ones next  – probably tomorrow – sorry, I’m just really tired :/.

Scholastic Goals:

1. get an A in English -> get straight A’s – god d*mn it, my overall GPA has been screwed to death because of my English grade ever since freshman year (consistent B’s). I have to get my act together on this one. I don’t know if my hopes for CalTech or Stanford are already screwed by the Bs, but hey, it won’t hurt to get an A. However, English has been my “kryptonite”, the bane of my high school existence. For some reason, it just doesn’t come to me as quickly (or as easily) as math or physics. Most people say I’m just not as interested in it, but I don’t know. But yeah, I HAVE to ace my English class – AP English Language and Comp, in order to get straight A’s.

2. win some golds in the Academic Decathlon – this is going to be my first year doing the Academic Decathlon (my schedule won’t let me do it the first 2 years of high school, for I have to get the required courses out of the way first). I’m going to try to get as many golds as I can get, but there’s a wide range of subjects, maybe I’m going to specialize on a few subjects. I still don’t know which ones, but definitely Super Quiz (history), Math, Science (chemistry), Economics.

3. Math Club becomes successful – hopefully I’m going to have a supportive cast of members (and moderator), but this thing has to be a success, at least for its first year of existence. I have to think about some clever ideas to even get some students join the club, but I’m not really worried about the sheer “size” of the club. I just have to make sure that the members I actually have are having a good time.

4. do the SAT – I’m going to take the actual SAT for the first time this year. I’ve done a few practice ones on the web, and my score ranged from 2250-2400. I have to push the lower boundary to about 2350 before I take the actual test, hopefully I can get the triple 800s.

5. SAT IIs + APs – I don’t know many I’m going to do, or how many I SHOULD do. I’m thinking about taking 5 tests – Math II, Physics, Chemistry, US History, Literature. I’m still iffy about taking the Spanish test, and whether I should take the one with listening or not.

And of course I have to do my best on the APs at the end of the year. I’ll be taking the Physics C: Mech, AP Calc BC, AP Lang and Comp., and AP US History. I would’ve preferred to take a few of these before, but my school won’t let me until I’ve actually taken the class. Unfortunately for me, I didn’t know of taking it at another location until someone had pointed that out to me recently :/

As you all know (or probably not), I’m in my junior year, the so-called “most critical” year by 75% or so by the American population (which definitely includes college admission committees), so this is going to be a tough year. Hopefully, the results I reap will be worth it.

Next post will be my personal goals – the books I’m going to read this year, and some other stuff. Stay tuned.

dent-murray

•September 6, 2009 • Leave a Comment

Totally is not what I’m expecting it to be. Dent had some highlights in the 1st set, but I think that’s as far as it goes, right now Murray leads him 2 sets to nil (6-3, 6-2). Now I hate that bastard even more.

On the other hand, hopefully Dent can use this run as a springboard to his full comeback to the game.

math club ideas + us open

•September 5, 2009 • Leave a Comment

So one of my friends and I decided that we were going to start the first math club in our high school. Now, I’m thinking of ideas to make it fun while we actually learn something interesting. I’ll have to go talk to the advisor next week for any suggestions, plus I have to drop off the AMC registration forms. Hopefully this math club thing can be somewhat succesful.

Meanwhile, I was just watching Day 6 of the U.S. Open tennis this morning. The first match the Hewitt-Federer. I actually thought that Hewitt was going to pull off the upset, after being up in the 3rd set 6-4, 3-6, 5-4. Roger was a little off during the first 3 sets. Then, he just turned into Roger Federer. Yeah..no need to say more. He beat Hewitt in 4 sets.

Now, I don’t know what to say about the other match. It was Novak Djokovic (woot) against some unknown fat*ss American Jesse Witten (no, seriously. Even the commentators said that he looked more like a TE than a tennis player). So anyways, the Djokers game looked REALLY off in this one, Witten won the 1st set and it looked like he was going to take another won. Luckily he had some rookie mistakes, and Novak took the next 3.

I don’t like how my boy Nole is playing as of late. He loses focus at times, either when he’s up by a respectable margin (either in a game or in a set), or when he is down. He doesn’t have the killer instinct he used to have. The reason why Federer was able to balance the scales against Hewitt was that he had a couple of big aces during the middle of the match, when it looked like Hewitt was going to pull away. I wish Djokovic would develop the same kind of coolness and collectiveness than Federer has. So anyways, I’m just glad that he was able to win the match, and he’s up against 15th seed Radek Stepanek next.

I’m looking forward to the Dent-Murray match. Taylor Dent has made quite a noise in his comeback run. Hopefully he can make a game out of this. And I hate Andy Murray. With a passion.

table lines and all that

•September 4, 2009 • 1 Comment

Today was nothing special, just a regular, normal school day (with the fall pep rally and all that crap). Although, I must confess, this day has not been one of my “good” days.

My AP Physics C teacher decided to dock points off my lab because of insignificant deviations from his preferred “norm” of a lab report. He takes off a few points off because my table (which any normal person would immediately recognize without any problem) did not have any line borders!! I mean, how ridiculous is that??

Plus, he takes off points because my sentence “structure” in my conclusion paragraph was not in the same format as the one he had in mind. My conclusion was perfectly the same as his (in terms of thought and meaning). But my “sentence 3″ (or so he says), should be technically before “sentence 2″.

It’s like this, say that the “sentence” he wanted was:

I scored more points, therefore I won the game.

The one I had was:

I won the game, because I scored more points.

He said he had to take points because “I scored more points” (in the above example) had to be before “I won the game”. When I heard that (compounded with the fact that he already took off points from my “borderless graph”) I was pissed more than anything else! >.<

This isn’t even English class, and my even English teacher won’t take off points off for such silly syntax “errors” (they’re not even errors). I didn’t want to argue with him anymore at that point, it was obvious what he was trying to imply.

Oh well, I guess that’s my story for today. Luckily, towards the end, he said he’d let us redo the whole lab report to make it “perfect” (like it had any mistakes to begin with). The rest of the day wasn’t better, especially when it’s freaking 105 degrees outside :/

initial ramblings

•September 3, 2009 • 1 Comment

Welcome to my blog. This is my first official post. I guess I made a blog so I can share my feelings and my journey through my life, especially in my passion in mathematics.

Mathematics is something that I treasure so much right now, and would probably treasure for the rest of my life. I think it’s a really interesting subject to say the least. It is the explanation different phenomena, from the inner workings of the tiniest quarks and leptons to the universal laws of relativity and motion. The abstraction behind all these equations is fascinating. However, where I am right now is just the beginning. There are so many things to be discovered and learned and to be shared with others.

Hopefully I can write some more, and share to you my experiences. One last thing I’d like to share with you is this powerful and ground-breaking equation discovered by Einstein in 1905:

e=mc^2

I can’t even begin to explain how amazing this equation is. Physicists have long sought an answer on how to express energy in terms of mass and motion. Different innovations have arisen as a result of this discovery, the most famous of which is the atomic bomb.

For next time, I shall write about different topics that I am currently working on, and some of the other things that fascinate me about the world of math and abstraction. I hope you’re ready, for I very well am.