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Thursday, December 15, 2005

Scribe for Today

So the scribe today is me! Well today in class, we had a surprise quiz...as usual :)

The first question was:

1) Which of the following examples of reasoning are INDUCTIVE and which are DEDUCTIVE?

Every time we have a club meeting, I have a test in school the next day.
REVIEW * Inductive is when we observe several particular examples of something and then conjecture that it must always be the case.
Susan’s father brought her early to school each day. She noticed that Ms. Taylor, her math teacher, arrived at 7:30 each day for several weeks. Susan said, “Ms. Taylor always arrives at 7:30.”
All students in senior high must enroll in physical education. John is a student, so he concludes that he will take physical education.
REVIEW * Deductive is when we argue from basic (inarguable) truths (like definitions) to a valid conclusion.
The sun has risen each morning from time immemorial. We can be certain it will rise tomorrow.
e. Anyone who likes to play football likes to play basketball. Sheeva likes to play football. We conclude that she likes to play basketball.
f. Triangle ABC is an equilateral triangle. We can conclude that AB is congruent to AC.
g. Joe counted the number of cars of different colors that passed his house in 15 minutes. More than half the cars were white. He decided that white is the most popular colors for cars.

The second question was:

2) Which of the above conclusions are VALID and which are INVALID?

REVIEW * Invalid is when its conclusion is NOT a direct consequence of its premises---even if its premises (basic truths) and conclusion (the result of an argument) are true.
REVIEW * Valid is when its result is a direct consequence of its premises.

After the quiz on logic, Mr. K gave us a Venn Diagram Problems sheet to work on the first problem which was called Music Survey and the problem was:

Recently several students at DMCI were questioned about their favorite music groups, with the following results:

- 22 like Hole
- 25 like U2
- 39 like Third Eye Blind
- 9 like U2 and Hole
- 17 like Hole and Third Eye Blind
- 20 like U2 and Third Eye Blind
- 6 like all three
- 4 like none of these performers

(a) How many students were questioned?

Also in class, Mr. K told us to write down the meaning of Counter Example on our dictionary. For those of you who missed the class today, here it is.

COUNTER EXAMPLES: given any logical argument of mathematical theorem, if you can find only one case where it is NOT true, then the entire argument of theorem is false and must be discarded---this is called a counter example.

Mr. K told us in class today about the “Fermat Theorem” which was really interesting and I even looked up on it on the internet to get more information about the Fermat Theorem. Here it is:

Pierre de Fermat died in 1665. Fermat is the most famous number theorist who ever lived. He was in fact a lawyer and an amateur mathematician. The fact that he published only one mathematical paper in his life and that was an anonymous article written as an appendix to a colleague's book. Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat's letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel written as a marginal note in his father's copy of Diophantus' s Arithmetica.
Fermat's Last Theorem states that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2.
Format wrote: “I have discovered a truly remarkable proof which this margin is too small to contain.” Fermat realized that his remarkable proof was wrong when he first studied Diophantus' s Arithmetica around 1630, however, since all his other theorems were stated and restated in challenge problems that Fermat sent to other mathematicians. Although the special cases of n = 3 and n = 4 were issued as challenges (and Fermat did know how to prove these) the general theorem was never mentioned again by Fermat.
In fact in all the mathematical work left by Fermat there is only one proof. Fermat proves that the area of a right triangle cannot be a square. Clearly this means that a rational triangle cannot be a rational square. In symbols, there do not exist integers x, y, z withx2 + y2 = z2 such that xy/2 is a square. From this it is easy to deduce the n = 4 case of Fermat's theorem.
It is worth noting that at this stage it remained to prove Fermat's Last Theorem for odd primes n only. For if there were integers x, y, z with xn + yn = zn then if n = pq, (xq)p + (yq)p = (zq)p.
Euler wrote to Goldbach on 4 August 1753 claiming he had a proof of Fermat's Theorem when n = 3. Euler's mistake is an interesting one, one which was to have a bearing on later developments. He needed to find cubes of the form p2 + 3q2 and Euler shows that, for any a, b if we put p = a3 - 9ab2, q = 3(a2b - b3) then p2 + 3q2 = (a2 + 3b2)3. The next major step forward was due to Sophie Germain. A special case says that if n and 2n + 1 are primes then xn + yn = zn implies that one of x, y, z is divisible by n. Hence Fermat's Last Theorem splits into two cases.
Case 1: None of x, y, z is divisible by n.
Case 2: One and only one of x, y, z is divisible by n.
Sophie Germain proved Case 1 of Fermat's Last Theorem for all n less than 100 and Legendre extended her methods to all numbers less than 197. At this stage Case 2 had not been proved for even n = 5 so it became clear that Case 2 was the one on which to concentrate. Now Case 2 for n = 5 itself splits into two. One of x, y, z is even and one is divisible by 5. Case 2(i) is when the number divisible by 5 is even; Case 2(ii) is when the even number and the one divisible by 5 are distinct.
Case 2(i) was proved by Dirichlet and presented to the Paris Académie des Science in July 1825. Legendre was able to prove Case 2(ii) and the complete proof for n = 5 were published in September 1825. In fact Dirichlet was able to complete his own proof of the n = 5 case with an argument for Case 2(ii) which was an extension of his own argument for Case 2(i).
In 1832 Dirichlet published a proof of Fermat's Last Theorem for n = 14. Of course he had been attempting to prove the n = 7 case but had proved a weaker result. The n = 7 case was finally solved by Lamé in 1839.He sketched a proof which involved factorizing xn + yn = zn into linear factors over the complex numbers. Lamé acknowledged that the idea was suggested to him by Liouville. However Liouville addressed the meeting after Lamé and suggested that the problem of this approach was that uniqueness of factorisation into primes was needed for these complex numbers and he doubted if it were true.
Much work was done in the following weeks in attempting to prove the uniqueness of factorization. Wantzel claimed to have proved it on 15 March but his argument He is correct about n = 2 (ordinary integers), n = 3 (the argument Euler got wrong) and n = 4 (which was proved by Gauss).]
Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved. It has the dubious distinction of being the theorem with the largest number of published false proofs. For example over 1000 false proofs were published between 1908 and 1912. The only positive progress seemed to be computing results which merely showed that any counter-example would be very large. Using techniques based on Kummer's work, Fermat's Last Theorem was proved true, with the help of computers, for n up to 4,000,000 by 1993.
Further work by other mathematicians showed that a counter-example to Fermat's Last Theorem would provide a counter -example to the Shimura-Taniyama-Weil Conjecture. The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA.
This, however, is not the end of the story. On 4 December 1993 Andrew Wiles made a statement in view of the speculation. He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However one problem remains and Wiles essentially withdrew his claim to have a proof. He states
The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semisquare case (of the symmetric square representation associated to a modular form) is not yet complete as it stands. I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures.
In March 1994 Faltings, writing in Scientific American, said
If it were easy, he would have solved it by now. Strictly speaking, it was not a proof when it was announced.
Weil, also in Scientific American, wrote
I believe he has had some good ideas in trying to construct the proof but the proof is not there. To some extent, proving Fermat's Theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest.
In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work.
Wiles worked on it for about two weeks, and then suddenly inspiration struck.
In a flash I saw that the thing that stopped it [the extension of Flach's method] working was something that would make another method I had tried previously work.
On 6 October Wiles sent the new proof to three colleagues including Faltings. All liked the new proof which was essentially simpler than the earlier one. Faltings sent a simplification of part of the proof. No proof of the complexity of this can easily be guaranteed to be correct, so a very small doubt will remain for some time. However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat's Last Theorem.

Click for the full information on Fermat Theorem



At 12/15/2005 11:43 PM, Blogger Mr. Kuropatwa said...

Cool scribe post! You might be interested in this; it will allow you to write your post entirely in Word then upload and post to blogger with the click of a button.

Also, I wonder if you can share the link to the site where you found all the excellent information about Fermat. Generally speaking, it's good blogger practice to always link to your sources. ;-)


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