<p>Thanks Senior0991. I hope the adcoms feel the same about me. And as for fizix2, I get your point. It is highly valid, and I think could serve a crushing blow to my proof. But I am slowly gathering my thoughts, and expect to reply soon. I might be addong a couple of things to show that the proof is valid only for natural numbers, but it may take a couple of days.</p>
<p>This is addressed to you mathboy98. As for refining my attitude towards maths, I don’t know how to quite put this, but I seem to have fallen in love with Maths and Science (for want of a better expression). I literally haunt the libraries around Singapore, reading up undergrad and grad maths and physics textbooks for fun. As you might have guessed, my social life is non-existent. Infact, I don’t quite remember the last time I touched my History and English books, but I still seem to do alright in them miraculously. I do stuff like this just as a serious hobby, and do not hope to accomplish anything or show the world what I am. I just came up with it after 2 weeks of struggling with the concepts involved, and have told you guys about it. So, as for refining my attitude towards maths, I know that I need to understand the comlpex machinery involved first, but this was just for fun; and most of the concepts used in my proof are not taught in high school.
Sadly, there are no theorems to prove in Physics.</p>
<p>Does satisfying the triangle inequality in all three directions necessarily imply that you have 3 valid sides to a triangle? I don’t remember this, and I’m not convinced.</p>
<p>Alright. Think of it this way. Let a, b and c be position vectors in the coordinate plane. If the triangle inequality is satisfied all three ways, a triangle can be formed. This is merely the converse of the triangle inequality, and brings number theory concepts to the coordinate plane.</p>
<p>I’m sorry. a, b and c can be any vectors, and not solely position vectors.</p>
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<p>Well, you can do a lot of good stuff without complex machinery, but my point is that I hope you spend your time, as you say, doing lots of reading. That’s what I did in high school, and what I’ve continued to do in college. I, much like you, read textbooks for fun, and sat in the front row of English class with a math book almost every day (and get this – I actually LIKE literature). My point was not even that you should go only after complex machinery, it’s that you should try to develop early on an idea of what it means to ask a <em>deep</em> mathematical question, because you’ll find that it’s not an easy thing to do. And I imagine an enthusiast like you will derive some pleasure from doing such a thing. </p>
<p>I applaud your enthusiasm, and would have loved to go to high school with you, as I’d have had someone else to talk to about the things I liked.</p>
<p>Why is it that [a^2 + b^2 - 2<em>a</em>b*cosC]^(n/2) cannot equal a^n + b^n where n is even and positive and C is given by the cosine rule? Given only the constraint that n must be positive and even, this statement is not true. For example, for a=1, b=3, cosC=(10-28^(2/3))/6, n = 4, then the above equations are equal.</p>
<p>This is what was pointed out earlier about needing to constrain from reals down to positive integers. I assume that your two contradictions are indeed impossible given that a and b are positive integers (because Fermat’s Last Theorem was proven), but you haven’t shown why they are impossible under this condition.</p>
<p>dLo. The example you have given above shows that c= 28^(1/3), while it has been explicitly stated at the beginning of the proof that a, b and c are (non-zero) natural numbers.</p>
<p>I think what jsd472’s point is that although it is about natural numbers, the proof also works for real numbers. So that’s a contradiction, as, for example, (3^(1/4))^4 + (4^(1/4))^4 = (7^(1/4))^4</p>
<p>And mathboy98, I would have loved to go to school with you too, and incidentally, I also love literature. :)</p>
<p>Hey. I’ve mailed you guys a more complete version of the proof, with proofs for the last assertions.</p>
<p>anyone there? :)</p>
<p>We do have other things to do. You can wait more than a day or two. Personally, if I were on admissions, this paper would hurt your chances. It depends on the person looking at it. True, it shows motivation and passion for math, which is good, but it also shows … just read my email.</p>
<p>yeah it does depend a lot on the person looking at it,
for example if i were on admissions, it would help your chances
especially if you didn’t try to claim it was correct and just said it was something you were trying to work out for fun but that had errors in it
so don’t be discouraged!
because seriously, what kind of nerd tries to prove fermat’s last theorem in his free time, that’s not even something i would do, you’re a pretty hardcore person
(unless you just did this to impress the admissions officers)</p>
<p>this is of course assuming that the admissions officers haven’t read this thread already…</p>
<p>thanks for your replys guys. jsd472, although a couple of your points could be refuted, the main point about divergent series is dead right. There were some pretty stupid mathematical manipulations towards the end, and you are not being blunt when you call it BS. Obviously the series does not converge to that value, because fermat’s last theorem has been proven. but proving the series are divergent is not the way to go about it. 
and fizix 2, thanks for having faith in me. i did not try to prove the theorem to impress the adcoms, because infact i attempted this problem in grade 11, when i did not even know that caltech existed. like most other science-oriented students, i wanted to go to MIT. and do the adcoms really check these forums? Well that is surprising!</p>
<p>caltech lets students sit on the admissions committee,
in fact ben golub was on the admissions committee
and he had thousands of posts</p>
<p>Pretty much every admissions committee is aware of this forum. Whether they choose to read it or not depends on the officer ;)</p>
<p>Oh. OK. I’m not quite sure what impression of myself did I create through this forum, but hopefully it is something good. So as to not misuse this forum to affect my chances of getting in, as I had initially just started this thread to discuss my mathematical fancies and to find out my chances of getting in (lame; I know), I will now wrap up this thread from my side. I’ll probably apply ED to Caltech (I’m not sure I am allowed to do so if I require financial aid), and my application may give a fuller picture of what I am all about. So the adcoms can expect to see “Ayush Khaitan” among the huge piles of applications tehy will recieve from the world over. I just really hope I have made a good impression on you guys, and just pray really, really, really hard to get in. Adios! :)</p>
<p>And thanks to all of you again for being so patient with me. Really appreciate that.</p>