A SPANISH LANGUAGE BIOGRAPHY OF PAUL ERDOS

In the first post of this new year 2014 I leave to my students in Argentina and to every Spanish language interested a new and original biography of the great Paul Erdos. It is written in Spanish but I will soon post it translated into English. Hope you enjoy it!

 

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This item is dedicated to my students of Mathematical Analysis 3 at Facultad de Ingenieria, University of Buenos Aires. First some background. We have seen in our first two classes how to solve quadratic and cubic equations with coefficients in C. The roots of the cubic equations are to be found by means of Tartaglia-Cardano' s formula. Because our course is not about algebra but about Analysis we will not study the solution of the quartic equation which is performed by means of Ludovico Ferrari's formula. We have also mentioned that there is no "algebraic formula" that gives the roots of the quintic, a fact proved by Galois, Abel and Ruffini. So, how can you be sure that any polinomial has a root in C if you can't effectively show it? The answer is the Fundamental Theorem of Algebra proved by Gauss in 1799 although its proof was completed by Ostrowski in 1920. The next proof is an existence non constructive one, which means that we will prove that there is a root but we will make no attempt to show which actually is it.

Fundamental Theorem of Algebra Let p(z) = a_n z^n + a_{n-1} z^{n-1} + \dots\ + a_1 z + a_0 be any polinomial with the a_{i} \in C and a_{n} \neq\ 0. Then there is a z_{0} \in C such that p(z_0) = 0.

Proof. We may assume p(z) real for z real. If this is not the case consider the polinomial q(z) = p(z) \overline{p}(z) (\overline{p}(z) being the same polynomial than p(z) but with its coefficients conjugated)   which is real for z real and the contradiction will be met by this last polinomial q(z) if not with the original p(z).

Since p(z) does not change sign because it has no roots then the integral :

\int_{0}^{2\pi} \frac{1}{p(2cos\theta)} d\theta

is either positive or negative but not zero.

If we introduce z = e^{(i\theta)} for 0 \leq \theta \leq 2\pi then this integral become :

\frac{1}{i}\oint \frac{1}{z p(z+z^{-1})} dz\frac{1}{i}\oint \frac{z^{n-1}}{z^{n} p(z+z^{-1})} dz.

Now, the integrand is analitic for z\neq 0 and for z = 0 we have that the denominator is a_{n} \neq 0.

Because the integrand is analytic, by Cauchy theorem the integral is zero. A contradiction. Thus, the Fundamental Theorem of Algebra is true.

Note : This beautiful proof was given by N.C.Ankeny, an american mathematician who was a fellow at Princeton and the Institute for Advance Studies. He specialized in Number Theory and he wrote a book on game theory and gambling.

 

A COMMENT FROM FAN

August 12th, 2013 | Posted by admin in Uncategorized - (0 Comments)

On August the 5 th I have posted on Tao's blog a link to my proof of the three consecutive non increasing inequalities between the differences of consecutive primes. This, I thought, was a question raised by Erdos and Turan in their 1948 paper "On some new questions on the distribution of prime numbers". On August 11 th Fan answer my post with the following information : Erdos and Turan were asking for decreasing inequalities (not non increasing ones). This information was already at the end of my work but Fan pointed out that with non decreasing inequality my results can be obtained from the prime number theorem. In the end the value of my work is the "elementary" proof of these non increasing inequalities without use of the prime number theorem. Thanks again to Fan.

In this post I don't intend to share one more Euler's biography. You can always find that on the internet. My objective in this section is to give the reader my opinion on what I think was the life of these mathematicians.

I was thinking a while on which mathematician should I choose first. And I concluded that Euler was the man.

First of all  I would like to say that he was the most prolific mathematician of all time. He wrote about 900 books! His theorems are to be found on almost all branches of math.

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But let's talk about his life. Leonhard Euler was the father of 13 children. He had to live the very bad moment of burying one of them. So in the end he was the father of "only" 12.Try to imagine a house full of kids and a wife. Where do you get the time and strength to do more than 2000 theorems? Nevertheless he made it.

At around age 30 he started to lose the right eye's sight. And at 50 the left one's. So he had to live part of his life as a blind man. But this fact did not imply less mathematics. To the contrary very notable contributions were found after he lost his second eye.

Euler, who never worked as a professor, had his doctoral thesis revised and approved by Johann Bernoulli. We owe Leonhard the following contributions :

THE NUMBERS e AND i, THE CONNECTION BETWEEN THE TRIGONOMETRIC FUNCTIONS AND THE EXPONENTIAL, THE F + V - E = 2 FORMULA FOR PLANAR GRAPHS FROM WHICH TOPOLOGY WAS BORN, THE EULERIAN INTEGRALS AMONG THEM THE GAMMA FUNCTION, THE EULER PHI FUNCTION OF NUMBER THEORY, THE APPROXIMATE SOLUTIONS OF DIFFERENTIAL EQUATIONS OF FIRST ORDER, THE FUNDAMENTAL EQUATION OF THE CALCULUS OF VARIATIONS, THE EXPRESSION OF THE EXPONENTIAL FUNCTION AS A LIMIT OF THE POWER (1+X/n)^n, THE PATHS ON EULERIAN GRAPHS, THE EQUATIONS OF CAUCHY-RIEMANN WERE ALREADY WELL KNOWN TO HIM, THE SAME WITH GREEN'S THEOREM, ETC, ETC, ETC.

On September 18th 1783 his soul left our world.

On April 15th 2013 he was featured at Google's homepage.

A comment from Gergely Harcos

July 12th, 2013 | Posted by admin in Opinion - (0 Comments)

On July the 10 th I have posted on Terry Tao 's webpage page my result on the three primes theorem. A few hours later Gergely Harcos from the Alfred Renyi institute pointed out that my result already existed. It was proved by Paul Erdos and Paul Turan in 1948. Here is the link of this remarkable paper, particularly inequalities (7).

http://www.ams.org/journals/bull/1948-54-04/S0002-9904-1948-09010-3/S0002-9904-1948-09010-3.pdf

When I published this theorem as mine in my website I was not aware that these great mathematicians had preceded me. But you may judge my good faith based on the fact that my proof is far simpler and is based on an entirely different construction.

 

As always I 'm not trying to compare myself with these great mathematicians by saying that my proof is simpler and based on an entirely different construction.

 

Thanks once again to Gergely Harcos.

 

Featured Mathematician

July 11th, 2013 | Posted by admin in Featured Mathematician - (0 Comments)

In this section I intend to talk about some famous mathematicians and to share interesting stories and facts.

Stay tuned.

My first post

July 11th, 2013 | Posted by admin in Opinion - (0 Comments)

My first post will be to explain why I have taken the work and time necessaries for this job.

I think that I can give my visitors a pure and sincere opinion of what I think and do about mathematics and I think I can help the very valuable students to gain faith and experience in their own researches and studies.

First of all I would like to say that the Three Primes Theorem that I have proved in the autumn of 2013 (South Hemisphere) is by no means a great theorem of Mathematics. It is not the Green Tao Theorem, nor the Twin Primes Conjecture and much less the Riemann Hypothesis.

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To the class and level of these great mathematicians I have not the slightest pretension to belong to.

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Nevertheless the theorem shares with them the fact that it is a result of the elusive prime numbers. Then it is a small grain of sand added to the Theory of Primes and thus to the science of Math.

And the second thing that I would like to say is that I am aware of each of the steps that my mind had taken in the construction and proof of this theorem. And it is on these steps that I shall write my next post.

This is (I think) the best I can give to my students : to explain them in detail these steps so they can try themselves to construct their own work.

Having this said I propose to take some time to write how I was able to prove this theorem.