Fermat’s last theorem – interesting observation

Today, while reading about an answer related Fermat’s last theorem in Quora, I came across an interesting comment that I felt was interesting.

It goes like this:

Fermat was mistaken. He could not possibly have devised a simple proof, The proof that was finally devised relied heavily on  a branch of Maths that wasnt even around at the time, known as Modular Functions.

Modular Form — from Wolfram MathWorld

There was originally an unsolved problem called the Taniyama–Shimura–Weil conjecture

Taniyama-Shimura Conjecture

which states that elliptic curves over the field of rational numbers are related to modular forms.  There was also a second unsolved problem, Ribets  theorem (earlier called the epsilon conjecture or ε-conjecture) which is a statement in number theory concerning properties of Galois representations associated with modular forms

Ribet’s Theorem — from Wolfram MathWorld

Frey Curve — from Wolfram MathWorld

In a nutshell:

Ribet  showed that Frey curves cannot be modular, so if the Taniyama-Shimura conjecture were true, Frey curves couldn’t exist and Fermat’s last theorem would follow with b even and a=-1 (mod 4). However, Frey did not actually prove that his curve was not modular. The conjecture that Frey’s curve was not modular came to be called the “epsilon conjecture,” and was quickly proved by Ribet (Ribet’s theorem) in 1986, establishing a very close link between two mathematical structures (the Taniyama-Shimura conjecture and Fermat’s last theorem) which appeared previously to be completely unrelated.

Only by solving these two problems was it realised that it also served as a proof for Fermats Last Theorem. The proof itself is over 150 pages long and consumed seven years of Wiles’s research time.

I wish to verify the authenticity of the statement above. What is more interesting is to understand what route Fermat might have possibly taken to answer this.

According to me, new findings in any research area help in aiding the long existing attempts to solve famous unsolved questions in that area. But lack of these cannot be a primary reason for the elusion of solution so far.

After long

I have been gone for long and also haven’t been having much time on my hands for writing new posts but starting today, I will try to change that and be more frequent in my blogging and present new articles form whatever I am reading and various other articles that interest me.

 

Recently I came across a particularly fascinating divisibility test for 7 in the blog of Tanya Khovanova who’s quite famous among the math blogging community.

 

Here is the post :http://blog.tanyakhovanova.com/?p=159

 

It basically gives you a graph that;s constructed by a certain David Wilson, a  fan of sequences who happened to be a guest blogger on the blog i mentioned above.

 

The main point of the test is as follows :

By David Wilson :

I have attached a picture of a graph.

Write down a number n. Start at the small white node at the bottom of the graph. For each digit d in n, follow d black arrows in a succession, and as you move from one digit to the next, follow 1 white arrow.

For example, if n = 325, follow 3 black arrows, then 1 white arrow, then 2 black arrows, then 1 white arrow, and finally 5 black arrows.

If you end up back at the white node, n is divisible by 7.

 

Now this made me look out for any other possible interesting ways to test the divisibility by 7 and that is ofcourse keeping in mind that the method i come across should be simpler then direct division process ;).

 

Another process that has been described in this paper is pretty simple nd staright forward but just a starting point in the author’s idea of divisibility by primes. Do give it a look if you are interested.

 

 

Galois’s 200th birthday

Évariste Galois was a French mathematician who died under mysterious circumstances after a duel when he was only 20 years old!
Even though he was so young, he is considered the father of modern algebra and he is the founder of Galois theory.
He was the first to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, the first to use the term group in mathematics in the modern way and so on.

 

In 1828, he attempted the entrance exam to École Polytechnique, the most prestigious institution for mathematics in France at the time, without the usual preparation in mathematics, and failed for lack of explanations on the oral examination. In that same year, he entered the École Normale (then known as l’École préparatoire), a far inferior institution for mathematical studies at that time, where he found some professors sympathetic to him.

In the following year, Galois’ first paper, on continued fractions,[3] was published. It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations. He submitted two papers on this topic to the Academy of Sciences. Augustin Louis Cauchy refereed these papers, but refused to accept them for publication for reasons that still remain unclear. However, in spite of many claims to the contrary, it appears that Cauchy recognized the importance of Galois’ work, and that he merely suggested combining the two papers into one in order to enter it in the competition for the Academy’s Grand Prize in Mathematics. Cauchy, a highly eminent mathematician of the time, considered Galois’ work to be a likely winner.

On July 28, 1829, Galois’ father committed suicide after a bitter political dispute with the village priest. A couple of days later, Galois made his second and last attempt at entering the Polytechnique, and failed yet again. It is undisputed that Galois was more than qualified; however, accounts differ on why he failed. The legend holds that he thought the exercise proposed to him by the examiner to be of no interest, and, in exasperation, threw at the examiner’s head the rag used to erase the blackboard. More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner, evoking the student’s rage. The recent death of his father may have also influenced his behavior.

Having been denied admission to the Polytechnique, Galois took the Baccalaureate examinations in order to enter the École Normale. He passed, receiving his degree on December 29, 1829. His examiner in mathematics reported, “This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research.”

In April 1829 Galois had his first mathematics paper published on continued fractions in the Annales de mathématiques. On 25 May and 1 June he submitted articles on the algebraic solution of equations to the Académie des Sciences. Cauchy was appointed as referee of Galois’ paper.

Galois sent Cauchy further work on the theory of equations, but then learned from Bulletin de Férussac of a posthumous article by Abel which overlapped with a part of his work. Galois then took Cauchy‘s advice and submitted a new article On the condition that an equation be soluble by radicals in February 1830. The paper was sent to Fourier, the secretary of the Paris Academy, to be considered for the Grand Prize in mathematics. Fourier died in April 1830 and Galois’ paper was never subsequently found and so never considered for the prize.

Galois, after reading Abel and Jacobi‘s work, worked on the theory of elliptic functions and abelian integrals. With support from Jacques Sturm, he published three papers in Bulletin de Férussac in April 1830. However, he learnt in June that the prize of the Academy would be awarded the Prize jointly to Abel (posthumously) and to Jacobi, his own work never having been considered.

There is a movie that is based on his life http://www.imdb.com/title/tt0205913/

In 1830, during the revolution, Galois was expelled from school for publicly criticizing the director of his school for failing to support the Revolution. Based on the suggestion of a friend Galois wrote a new paper on his research though. “Sur les conditions de re’solubilite’ des e’quations par radicaux,” is the only finished article on his theory of the solution of equations. Unfortunately, the Academy returned his paper stating that he needed to write a fuller explanation. Shortly after being expelled from school, Galois was arrested for political offenses and spent most of the last year and a half of his life in prison. He did write a scratchy and hastily written account of his researches which he entrusted to his friend August Chevalier. This account was written the night before his death and has been preserved. He was killed in a dual on May 31, 1832. The first full and clear presentation of Galois theory was given in 1870 by Camille Jordan in a book.

A famous quote from Galois:

“Unfortunately what is little recognized is that the most worthwhile scientific books are those in which the author clearly indicates what he does not know; for an author most hurts his readers by concealing his difficulties.”

More on his work in subsequent post

 

 

Hilbert’s 23 problems

Found this over here and sharing it over here

 

Hilbert’s address of 1900 to the International Congress of Mathematicians in Paris is perhaps the most influential speech ever given to mathematicians, given by a mathematician, or given about mathematics. In it, Hilbert outlined 23 major mathematical problems to be studied in the coming century. Some are broad, such as the axiomatization of physics (problem 6) and might never be considered completed. Others, such as problem 3, were much more specific and solved quickly. Some were resolved contrary to Hilbert’s expectations, as the continuum hypothesis (problem 1).

Hilbert’s address was more than a collection of problems. It outlined his philosophy of mathematics and proposed problems important to his philosophy.

Although almost a century old, Hilbert’s address is still important and should be read (at least in part) by anyone interested in pursuing research in mathematics.

In 1974 a symposium was held at Northern Illinois University on the Mathematical developments arising from Hilbert problems. A major mathematician discussed progress on each problem and how work on the problem has influenced mathematics. Also, 23 new problems of importance were described. The two-volume proceedings of the symposium was edited by Felix Browder and published by the American mathematical Society in 1976. See also Irving Kaplansky’s Hilbert’s problems, University of Chicago, Chicago, 1977.

There is also a collection on Hilbert’s Problems, edited by P. S. Alexandrov, Nauka, Moscow, 1969, in Russian, which has been translated into German.

Types of Infinities

One surely would have heard the word infinity many times in his life, but what most of the people dont know is that there are different kinds of infinities.

This post tries to throw some light on this.

You can see that there are infinite types of infinity via Cantor’s theorem which shows that given a set A, its power set P(A) is strictly larger in terms of infinite size (the technical term is “cardinality”).

http://en.wikipedia.org/wiki/Cantor%27s_theorem

A very nice introduction to the many different notions of infinity in mathematics is Rudy Rucker’s book: Infinity and the Mind. Unlike many other popularizations, this is written by someone who did a Ph.D. on the topic. Moreover, Rucker has gone to great lengths to make the presentation faithful to the mathematics but still accessible to an educated layperson. Below is an excerpt on the Alephs.

Now coming to the concept of countable and uncountable infinities, here’s an explanation that i have found really interesting:

Suppose no one every taught you the names for ordinary numbers. Then suppose that you and I agreed that we would trade one bushel of corn for each of my sheep. But there’s a problem, we don’t know how to count the bushels or the sheep! So what do we do?

We form a “bijection” between the two sets. That’s just fancy language for saying you pair things up by putting one bushel next to each of the sheep. When we’re done we swap. We’ve just proved that the number of sheep is the same as the number of bushels without actually counting.

We can try doing the same thing with infinite sets. So suppose you have the set of positive numbers and I have the set of rational numbers and you want to trade me one positive number for each of my rationals. Can you do so in a way that gets all of my rational numbers?

Perhaps surprisingly the answer is yes! You make the rational numbers into a big square grid with the numerator and denominators as the two coordinates. Then you start placing your “bushels” along diagonals of increasing size, see wikipedia.

This says that the rational numbers are “countable” that is you can find a clever way to count them off in the above fashion.

The remarkable fact is that for the real numbers there’s no way at all to count them off in this way. No matter how clever you are you won’t be able to scam me out of all of my real numbers by placing a natural number next to each of them. The proof of that is Cantor’s clever “diagonal argument.”
For those interested in the origin of this above discussion can go here: http://math.stackexchange.com/questions/1/different-kinds-of-infinities/9#9

More on this in my next post.