## martes, 23 de abril de 2013

### An oft-cited comment of Mr. Darwin

"During the three years which I spent at Cambridge my time was wasted, as far as the academical studies were concerned, as completely as at Edinburgh and at school. I attempted mathematics, and even went during the summer of 1828 with a private tutor (a very dull man) to Barmouth, but I got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense..."

Fuente: The Autobiography of Charles Darwin... Me enteré del origen de esta cita a través de un artículo reciente de D. H. Bailey y J. M. Borwein en la sección de ciencia de The Huffington Post (Why E. O. Wilson is wrong... [Fecha de publicación: 04/17/2013, 4:34 P.M.]). Al momento que escribo esto, lo único que encuentro con respecto a esta cita en la entrada en inglés de Wikiquote sobre Charles Darwin es lo siguiente:

Mathematics seems to endow one with something like a new sense.

Dedicamos esta entrada a todas aquellas personas para las que la precisión en este tipo de asuntos es esencial...

## miércoles, 10 de abril de 2013

### I. Tamm and the remainder term in Taylor's theorem

George Gamow, the physicist... who escaped to the United States from Stalinist Russia, tells the following tale of what can befall an innocent scholar in times of political turbulence.

Here is a story told to me by one of my friends who was at that time a young professor of physics in Odessa. His name was Igor Tamm (Nobel Prize Laureate in Physics, 1958). Once when he arrived in a neighbouring village, at the period when Odessa was occupied by the Reds, and was negotiating with a villager as to how many chickens he could get for half a dozen silver spoons, the village was captured by one of the Makhno bands, who were roaming the country, harassing the Reds. Seeing his city clothes (or what was left of them), the capturers brought him to the Ataman, a bearded fellow in a tall black fur hat with machine-gun cartridge ribbons crossed on his broad chest and a couple of hand grenades hanging on the belt.

'You son-of-a-bitch, you Communistic agitator, undermining our mother Ukraine! The punishment is death'.

'But no', answered Tamm. 'I am a professor at the University of Odessa and have come here only to get some food.'

'Rubbish!', retorted the leader. 'What kind of professor are you?'

'I teach mathematics'.

'Mathematics?', said the Ataman. 'All right! Then give me an estimate of the error one makes by cutting off Maclaurin's series at the nth term. Do this and you will go free. Fail and you will be shot!'

Tamm could not believe his ears, since this problem belongs to a rather special branch of higher mathematics. With a shaking hand, and under the muzzle of the gun, he managed to work out the solution and handed it to the Ataman.

'Correct!', said the Ataman. 'Now I see that you really are a professor. Go home!'

Who was this man? No one will ever know. If he was not killed later on, he may well be lecturing now on higher mathematics in some Ukrainian university.

W. Gratzer, Eurekas and euphorias: the Oxford book of scientific anecdotes. Oxford University Press 2002, pág. 36.

## lunes, 18 de marzo de 2013

### De Siegel a Mordell

Göttingen, March 3, 1964.

Dear Professor Mordell,

Thank you for the copy of your review of Lang's book. When I first saw this book, about a year ago, I was disgusted with the way in which my own contributions to the subject had been disfigured and made unintelligible. My feeling is very well expressed when you mention Rip van Winkle!

The whole style of the author contradicts the sense for simplicity and honesty which we admire in the works of the masters in number theory—Lagrange, Gauss or, on a smaller scale, Hardy, Landau. Just now Lang has published another book on algebraic numbers which, in my opinion, is still worse than the former one. I see a pig broken into [sic has broken into] a beautiful garden and rooting up [sic started rooting up] all flowers and trees.

Unfortunately there are many "fellow-travelers" who have already disgraced a large part of algebra and function theory; however, until now, number theory had not been touched. These people remind me of the impudent behaviour of the national socialists who sang: "Wir werden weiter marschieren, bis alles in Scherben zerfällt!"

I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction—I call it: Theory of the empty set—cannot be blocked up. Let us hope that your review may be helpful.

I still remember the nice time we had together during your visit in Göttingen.

With best wishes, also to Mrs. Mordell,

Carl Siegel.

[De acuerdo con lo que se puede leer en la cuenta de David C. Kandathil en Google+, él transcribió la carta de una edición facsimilar de la misma que se encuentra en los archivos del College of St. John de la Universidad de Cambridge.]

## lunes, 11 de marzo de 2013

### Apuntes para una historia del problema de los NPI

«... In 1985, a high school senior in Hawaii named David Williamson, who was taking mathematics courses at the University of Honolulu, proved that if an odd perfect number exists, it must have exactly one prime factor that, when divided by 4, leaves a remainder of 1. Williamson's professor didn't know whether the result was original and suggested he write to the legendary Erdös, who had just come through town. Williamson eventually got a letter back from Erdös: 'The result you proved is in fact due to Euler. He also proved that every even perfect number is of the form 2p-1(2p-1) where 2p-1 is a prime. It is also known (proved by Carl Pomerance) that an odd perfect number if it exists must have at least 7 distinct prime factors. Perhaps the following problem of mine will interest you...' Williamson, now a combinatorialist at IBM, was thrilled. 'This letter to a high school student won't rank very high on Erdös's list of accomplishments, but it did mean a lot to me.'»

Paul Hoffman, The man who loved only numbers: the story of Paul Erdös and the search for mathematical truth. Hyperion, NY, 1998, pp. 47-48.

## miércoles, 27 de febrero de 2013

### Excerpts on the genetic approach

1. «... The second premise for these notes is that in order for an introduction to differential geometry to expose the geometric aspect of the subject, an historical approach is necessary... Of course, I do not think that one should follow all the intricacies of the historical process, with its inevitable duplications and false leads. What is intended, rather, is a presentation of the subject along the lines which its development might have followed; as Bernard Morin said to me, there is no reason, in mathematics any more than in biology, why ontogeny must recapitulate philogeny. When modern terminology is introduced, it should be as an outgrowth of this (mythical) historical development.»

Mike Spivak, from the preface to the first edition of volume one of his A comprehensive introduction to Differential Geometry.

2. «... The basic method of the book is...the genetic method. The dictionary defines the genetic method as "the explanation or evaluation of a thing or event in terms of its origin and development."... It is important to distinguish the genetic method from history. The distinction lies in the fact that the genetic method primarily concerns itself with the subject—its "explanation or evaluation" in the definition above—whereas the primary concern of history is an accurate record of the men, ideas, and events which played a part in the evolution of the subject. In a history there is no place for detailed descriptions of the theory unless it is essential to an understanding of the events. In the genetic method there is no place for a careful study of the events unless it contributes to the appreciation of the subject. This means that the genetic method tends to present the historical record from a false perspective. Questions which were never successfully resolved are ignored. Ideas which led into into blind alleys are not pursued. Months of fruitless effort are passed over in silence and mountains of exploratory calculations are dispensed with. In order to get to the really fruitful ideas, one pretends that human reason moves in straight lines from problems to solutions. I want to emphasize as strongly as I can that this notion that reason moves in straight lines is an outrageous fiction which should not for a moment be taken seriously. Samuel Johnson once said of the writing of biography that "If nothing but the bright side of characters should be shown, we should sit down in despondency, and think it utterly impossible to imitate them in anything. The sacred writers related the vicious as well as the virtuous actions of men; which had this moral effect, that it kept mankind from despair." This book does, for the most part, show only the bright side, only the ideas that work, only the guesses that are correct. You should bear in mind that this is not a history or biography and you should not despair. You may well be interested less in the contrast between history and the genetic method than in the contrast between the genetic method and the more usual method of mathematical exposition. As the mathematician Otto Toeplitz described it, the essence of the genetic method is to look to the historical origins of an idea in order to find the best way to motivate it, to study the context in which the originator of the idea was working in order to find the "burning question" which he was striving to answer... In contrast to this, the more usual method pays no attention to the questions and presents only the answers. From a logical point of view only the answers are needed, but from a psychological point of view, learning the answers without knowing the questions is so difficult that it is almost impossible. That, at any rate, is my own experience...»

H. M. Edwards, from the preface to his Fermat's Last Theorem (A genetic introduction to algebraic number theory).

## sábado, 23 de febrero de 2013

### Excerpts

1. «For all of Kummer's work on the cutting edge of number theory, he was apparently rather bad at elementary arithmetic. One story has him standing before a blackboard trying to compute 7 times 9. "Ah," Kummer said to his high school class, "7 times 9 is eh, uh, is uh..." "61," one of his students volunteered. "Good," said Kummer, and wrote 61 on te board. "No," said another student, "it's 69." "Come, come, gentlemen," said Kummer, "it can't be both. It must be one or the other." (Erdös liked to tell another version of how Kummer computed 7 times 9: "Kummer said to himself, 'Hmmm, the product can't be 61 because 61 is a prime, it can't be 65 because that's a multiple of 5, 67 is a prime, 69 is too big—that leaves only 63.'")»

Paul Hoffman, The man who loved only numbers: the story of Paul Erdös and the search for mathematical truth. Hyperion, NY, 1998, pp. 208-209.

2. "… The propagation of unchecked and uncheckable anecdotes about the history of mathematics is a form of pollution to be combatted. An occasional tall tale, with appropriate caveats, can certainly be used to spice up the exposition from time to time, but when no sources are given for anything, such tales become an unacceptable norm."

H. M. Edwards, Review of three popular books on the Riemann Hypothesis. Math. Intelligencer. 26 1 (2004), p. 57.

## jueves, 4 de octubre de 2012

### Números poderosos y la conjetura abc

Consideremos la siguiente

Definición. Sea $n$ un número natural. Decimos que $n$ es un número poderoso si para cada primo positivo $p$ tal que $p|n$ se cumple que $p^{2}|n$.

No es difícil generar (infinitos) ejemplos de números naturales poderosos. Un problema clásico relacionado con estos números pide determinar si hay infinitos pares de números poderosos consecutivos. Un par de tal índole es $(8,9)$, otro es $(288,289)$ y una manera rápida de mostrar que la respuesta al problema es afirmativa es considerando la ecuación diofántica $x^{2}-8y^{2}=1$. Claramente, toda solución $(x,y)$ a la ecuación da lugar a un par de números poderosos consecutivos. Como la ecuación es Pell entonces admite una infinidad de soluciones en enteros y estamos... La solución obtenida da pie a otras cuestiones (como siempre, un problema por cada solución), pero prefiero concentrarme por ahora en el análogo del problema original para tres y cuatro números poderosos consecutivos. Que no puede haber cuatro números poderosos consecutivos es fácil de establecer pues entre cuatro números consecutivos siempre hay uno que es el doble de un número impar; es claro que tal número no puede ser poderoso. En el otro caso (tres números poderosos consecutivos), el dato indica que a la fecha no se conoce una sola tripleta de números poderosos consecutivos. Se conjetura, de hecho, que no hay tripletas de tal índole. Interesante: hay infinitos pares de númerosos poderosos consecutivos, no puede haber cuatro números poderosos consecutivos y no se sabe si existe siquiera una tripleta de números poderosos consecutivos. Un fenómeno similar ocurre en el estudio de números primos gemelos (números primos cuya diferencia es $\pm 2$). Aunque no se sabe si hay infinitos pares de números primos gemelos (se cree que sí), es muy fácil probar que sólo hay dos $3$-adas $(a,b,c)$ de números primos tales que $c-b=2=b-a$.

En fin, sacamos a relucir el tema de las tripletas de números poderosos consecutivos a consecuencia de la nota de hace algunas semanas en torno al Profesor Shinichi Mochizuki, sus investigaciones y la

Conjetura abc: [Denotemos con $N_{0}(k)$ al radical de $k$ (el producto de los primos positivos que dividen a $k$).] Para cada $\epsilon > 0$, existe $C(\epsilon)$ tal que si $a, b$ y $c$ son enteros distintos de cero y coprimos que cumplen que $a+b=c$ entonces

$\begin{eqnarray}\max\{|a|,|b|,|c|\} \leq C(\epsilon)\cdot N_{0}(abc)^{1+\epsilon}.\end{eqnarray}$

Hay varios problemas interesantes en Aritmética que pueden derivarse (condicionalmente, por ahora) de la conjetura abc. Uno de los ejemplos más notables es una versión asintótica del último "teorema" de Fermat... Otro ejemplo sería una versión débil del problema de las tripletas de números poderosos consecutivos; a saber, sólo hay un número finito de tripletas de números poderosos consecutivos. Mostraremos a continuación como ambos resultados saldrían de la conjetura abc.

Para el último "teorema" de Fermat, empezamos por notar que si para cierto $n \in \{3, 4, 5, 6, \ldots\}$ hay una solución en enteros (distintos de cero) a la ecuación $\begin{eqnarray*}x^{n}+y^{n}=z^{n}\end{eqnarray*}$, entonces hay una solución en enteros distintos de cero y coprimos. Luego, si la conjetura abc es cierta, existe una constante $C$ tal que

$|x^{n}| \leq CN_{0}(xyz)^{2}\leq C(xyz)^{2}$,

$|y^{n}| \leq CN_{0}(xyz)^{2}\leq C(xyz)^{2}$

y

$|z^{n}| \leq CN_{0}(xyz)^{2}\leq C(xyz)^{2}$.

Ergo,

$|xyz|^{n} \leq C^{3}|xyz|^{6}.$

De la desigualdad anterior se colige que $n$ está acotado superiomente. En otras palabras, el último "teorema" de Fermat es cierto para todo exponente suficientemente grande. La determinación explícita de un umbral para los exponentes a partir del cual valdría el último "teorema" de Fermat dependería de que tan explícitas puedan hacerse las constantes $C(\epsilon)$.

Analicemos ahora el problema de las tripletas de números poderosos consecutivos. Supongamos que $n-1$, $n$ y $n+1$ son números poderosos. Puesto que el conjunto de números poderosos es cerrado bajo el producto usual de $\mathbb{Z}$ y el radical de todo número poderoso es menor o igual a su raíz cuadrada, al considerar la igualdad $(n^{2}-1)+1=n^{2}$, la conjetura abc implica la existencia de una constante $C$ tal que
\begin{eqnarray*}n^{2} &\leq& CN_{0}((n^{2}-1)n^{2})^{1+1/4}\\ &=& CN_{0}((n-1)(n+1))^{1+1/4}N_{0}(n)^{1+1/4}\\ &\leq& C \left(\sqrt{(n-1)n(n+1)}\right)^{1+1/4}\\ &<& C(n^{3/2})^{1+1/4}\\ &=& C n^{15/8}.\end{eqnarray*} De esto se desprende inmediatamente que $n$ está acotado superiomente. Por consiguiente, si la conjetura abc es cierta, el número de tripletas de números poderosos consecutivos es finito...

No recuerdo bien si en el libro de Singh se menciona la conjetura abc, lo que no olvido es la vez que un profesor la mencionó cuando le comenté de un quickie que acababa de idearme:

68. ¿Existe una tripleta $\{A,B,C\}$ de matrices $m\times m$, $n \times n$ y $p \times p$, respectivamente, con entradas racionales tales que: 1) $m$, $n$ y $p$ son números naturales mayores que uno, 2) ninguna de las matrices tiene al cero como valor propio y 3) $(\mathbf{p}_{A}(t))^{2012}+(\mathbf{p}_{B}(t))^{2012}=(\mathbf{p}_{C}(t))^{2012},$ donde $\mathbf{p}_{D}(t)$ denota al polinomio característico de la matriz $D$?

Obviamente, en lugar de $2012$ el exponente en aquella ocasión era el número del año correspondiente. La conexión es con el origen mismo de la conjetura abc. Vea, por ejemplo, las páginas 38 y 39 del artículo Old and new conjectured diophantine inequalities del Profesor Serge Lang (Bull. Amer. Math. Soc. 23 1 (1990), págs. 37-75.) o ponga "teorema de Mason-Stothers" en Google.

¡Hasta pronto!

Referencias

[1] S. W. Golomb. Powerful numbers. Amer. Math. Monthly 77 8 (1970), págs. 848-852.
[2] M. B. Nathanson. Elementary methods in number theory. Springer Verlag, NY, 2000.