miércoles, 26 de abril de 2017

A play on the interplay 'twixt primes and polynomials

In what follows, we shall denote the set of positive prime numbers by $\mathbf{P}$.

Act I. It is more or less well-known that there does not exist a non-constant polynomial $f \in \mathbb{Z}[x]$ such that $f(n) \in \mathbf{P}$ for every $n \in \mathbb{N}$. This can be proven by reductio ad absurdum: if $f(n) \in \mathbf{P}$ for every $n \in \mathbb{N}$ and $f(1) =: p$, then $p \mid f(1+kp)$ for every $k \in \mathbb{N}$; it follows that at least one of the equations $f(x)=p$ or $f(x)=-p$ has more solutions than $\deg(f)$, Q. E. A. This result is typically attributed to Christian Goldbach: W. Narkiewicz, on page 25 of his "The Development of Prime Number Theory", even mentions that it can be found in a letter from Goldbach to Euler written on September 28th, 1743. Luckily for us, Springer-Verlag published two years ago a translation into English of the correspondence of L. Euler with C. Goldbach edited and commented by F. Lemmermeyer and M. Mattmüller.

Act II. Several years ago, while perusing a very interesting article on primes in arithmetic progressions by M. R. Murty, I learned the notion of prime divisor of a polynomial: if $p \in \mathbf{P}$ and $f \in \mathbb{Z}[x]$, we say that $p$ is a prime divisor of $f$ if $p \mid f(n)$ for some $n \in \mathbb{Z}$. According to Murty, the basic theorem on the set of prime divisors of a non-constant $f \in \mathbb{Z}[x]$ can be traced back (at least) to a 1912 paper of I. Schur. The theorem can be proven emulating the celebrated proof of Eucl. IX-20.

Theorem. If $f$ is a non-constant polynomial of integer coefficients, then its set of prime divisors is infinite.

Proof. If $f(0)=0$, then every $p \in \mathbf{P}$ is a prime divisor of $f$. If $f(0) = c \neq 0$, then $f$ has at least one prime divisor as it can take on the values $\pm 1$ only finitely many times. Given any finite set $\mathcal{P}_{k}:=\{p_{1}, \ldots, p_{k}\}$ of prime divisors of $f$, we are to show that we can always find another prime divisor of $f$ which does not belong to $\mathcal{P}_{k}$. Let $A := p_{1} \cdots p_{k}$ and consider the equality $f(Acx)=cg(x)$ where $g \in \mathbb{Z}[x]$ is a polynomial of the form $1+c_{1}x+c_{2}x^{2}+\cdots$ where every $c_{i}$ is a multiple of $A$. Since $g$ is a non-constant polynomial whose constant term is different from $0$, $g$ has at least one prime divisor $p$. Clearly enough, this prime number $p$ is also prime divisor of $f$ which does not belong to $\mathcal{P}_{k}$. Q.E.D.

Act III. Resorting to the ideas in the previous paragraphs plus Dirichlet's glorious theorem on primes in arithmetic progressions, we are going to determine all the non-constant polynomials $f \in \mathbb{Z}[x]$ such that $f(\mathbf{P}) \subseteq \mathbf{P}$.

- If $f$ is one such polynomial and $f(0)=0$, then $f(x)=xg(x)$ for some some $g \in \mathbb{Z}[x]$. Given that $p \cdot g(p) =f(p) \in \mathbf{P}$ for every $p \in \mathbf{P}$, it follows that $g(p) = \pm 1$ for every $p \in \mathbf{P}$; therefore, in this case we obtain that either $g(x)=1$ and $f(x) =x$ or $g(x)=-1$ and $f(x)=-x$.
- Let us assume now that $f$ is one such polynomial and $f(0)=c \neq 0$. By the above theorem, we may fix a prime divisor $q$ of $f$ which is greater than $|c|$. If $n \in \mathbb{Z}$ is a witness of the fact that $q$ is a prime divisor of $f$, then $q \nmid n$. Thus, if $p_{1} < p_{2} < p_{3} < \ldots$ are all the positive primes in the arithmetic progression whose first term is $n$ and whose common difference is $q$, we have that $f(p_{i}) \equiv f(n) \equiv 0 \pmod{q}$ for every $i \in \mathbb{N}$, which is decidedly absurd because $f$ can assume the values $\pm q$ only finitely many times.

Hence, $f(x)=x$ is the only non-constant polynomial with integer coefficients which maps sends $\mathbf{P}$ to one of its subsets.


martes, 18 de abril de 2017

Nunca hubo milagro


Banach y Tarski se encontraban gesticulando y argumentando, en el mismo cubículo, frente a un inmenso pizarrón verde, cuando demostraban el teorema que sería conocido a la postre como la Paradoja de Banach-Tarski: dada una bola sólida en $\mathbb{R}^{3}$, existe una descomposición de esta en un número finito de subconjuntos disjuntos que se pueden juntar otramente para producir dos copias idénticas a la bola original. Justo cuando terminaron la prueba, ambos callaron y se miraron muy contentos. Tarski hizo una pequeña aspiración y retuvo el aire un instante hasta que finalmente, abstraído, le dijo a Banach: "Ahora sabemos cómo fue que Cristo multiplicó los peces y el pan".

Autor: Enrique Ruiz.

Postdata. Leí este cuento por vez primera en 2016: no obstante, debo de confesar que estuve aguardando su aparición desde aproximadamente el primer semestre de 2004 pues fue más o menos por aquel tiempo que el Prof. Vulfrano T. me comentó que la multiplicación de los panes y los peces se podía conectar con el Axioma de Elección.

sábado, 7 de enero de 2017

Para la reflexión

«It is related of the Socratic philosopher Aristippus (c. 435 – c. 356 BCE) that, being shipwrecked and cast ashore on the coast of the Rhodians, he observed geometrical figures drawn thereon, and cried out to his companions: "Let us be of good cheer, for I see the traces of man." With that he made for the city of Rhodes, and went straight to the gymnasium. There he fell to discussing philosophical subjects, and presents were bestowed upon him, so that he could not only fit himself out, but could also provide those who accompanied him with clothing and all other necessaries of life. When his companions wished to return to their country, and asked him what message he wished them to carry home, he bade them say this: that children ought to be provided with property and resources of a kind that could swim with them even out of a shipwreck...»

(Vitruvio en De architectura [Libro VI])

viernes, 14 de octubre de 2016

Yet another function-theoretic proof of the Fundamental Theorem of Algebra

Let $f$ be a nonconstant polynomial with complex coefficients. Since $|f(z)| \to \infty$ as $z \to \infty$, we guarantee the existence of $R>0$ such that $$|f(z)|>|f(0)| \quad \quad (\ast)$$ for every $z \in \mathbb{C} \setminus \mathrm{B}_{R}(0)$. On the other hand, the continuity of the function $F \colon \overline{\mathrm{B}_{R}(0)} \to \mathbb{C}$ given by $z \overset{F}{\longmapsto} |f(z)|$ and the compactness of $\overline{\mathrm{B}_{R}(0)}$ allow us to ascertain the existence of $z_{0} \in \overline{\mathrm{B}_{R}(0)}$ such that $$|f(z_{0})| \leq |f(z)|$$ for every $z \in \overline{\mathrm{B}_{R}(0)}$. From $(\ast)$ we infer that $z_{0}$ is actually an element of $\mathrm{B}_{R}(0)$; then, by resorting to the Minimum-Modulus Principle, we conclude that $|f(z_{0})|$ must be equal to $0$ and we are done.

Scholia. a) If I understand correctly, the basic idea in this approach to the Fundamental Theorem of Algebra can be traced back to a 1748 memoir of d' Alembert. Yet, according to what we read in Reinhold Remmert's essay on the Fundamental Theorem of Algebra in [1, pp. 99-122], there were some gaps in d' Alembert's argument therein that would be pointed out by a twenty-two-year-old Gauss in the beginning of his doctoral thesis "Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse", which he submitted to Pfaff at the University of Helmstedt in 1799 and through which he obtained his doctorate. However, it is noteworthy that, on that occasion, "... Gauss also [remarked], almost prophetically (Werke 3, p.11): 'For these reasons I am unable to regard the proof by d' Alembert as entirely satisfactory, but that does not prevent, in my opinion, the essential idea of the proof from being unaffected, despite all objections; I believe that ... a rigorous proof could be constructed on the same basis.'"
b) Interestingly enough, the proof of the Fundamental Theorem of Algebra showcased by Aigner & Ziegler's in their Proofs from THE BOOK (5th. edition, pp. 147-149) is based on the aforementioned d'Alembertian attack as subsequently simplified by Argand in 1814.

[1] H. D. Ebbinghaus, et al., Numbers. Graduate Texts in Mathematics 123, Springer-Verlag, NY, 1991.

jueves, 21 de mayo de 2015

Some very interesting paragraphs on the axiomatic method and its connection to some glorious moments in American History

«In the time of Euclid, and for over two thousand years thereafter, the postulates of geometry were thought of as self-evident truths about physical space; and geometry was thought of as a kind of purely deductive physics. Starting with the truths that were self-evident, geometers considered that they were deducing other and more obscure truths without the possibility of error. (Here, of course, we are not counting the casual errors of individuals, which in mathematics are nearly always corrected rather promptly.) This conception of the enterprise in which geometers were engaged appeared to rest on firmer and firmer ground as the centuries wore on. As the other sciences developed, it became plain that in their earlier stages they had fallen into fundamental errors. Meanwhile the "self-evident truths" of geometry continued to look like truths, and also continued to seem self-evident.

With the development of hyperbolic geometry, however, this view became untenable. We then had two different, and mutually incompatible, systems of geometry. Each of them was mathematically self-consistent, and each of them was compatible with our observations of the physical world. From this point on, the whole discussion of the relation between geometry and physical space was carried on in quite different terms. We now think not of a unique, physically "true" geometry, but of a number of mathematical geometries, each of which may be a good or bad approximation of physical space, and each of which may be useful in various physical investigations. Thus we have lost our faith not only in the idea that simple and fundamental truths can be relied upon to be self-evident, but also in the idea that geometry is an aspect of physics.

This philosophical revolution is reflected, oddly enough, in the differences between the early passages of the Declaration of Independence and the Gettysburg Address. Thomas Jefferson1 wrote:

"... We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness..."

The spirit of these remarks is Euclidean. From his postulates, Jefferson went on to deduce a nontrivial theorem, to the effect that the American Colonies had the right to establish their independence by force of arms.

Lincoln spoke in a quite different style:

"Fourscore and seven years ago our fathers brought forth on this continent a new nation, conceived in liberty and dedicated to the proposition that all men are created equal."

Here Lincoln is referring to one of the propositions mentioned by Jefferson, but he is not claiming, as Jefferson did, that this proposition is self-evidently true, or even that it is true at all. He refers to it merely as a proposition to which a certain nation was dedicated. Thus, to Lincoln, this proposition is a description of a certain aspect of the United States (and, of course, an aspect of himself). (I am indebted for this observation to Lipman Bers.)

This is not to say that Lincoln was a reader of Lobachevsky, [János] Bolyai or Gauss, or that he was influenced, even at several removes, by people who were. It seems more likely that a shift in philosophy had been developing independently of the mathematicians, and that this helped to give mathematicians the courage to undertake non-Euclidean investigations and publish the results.

At any rate, modern mathematicians use postulates in the spirit of Lincoln. The question whether the postulates are "true" does not even arise. Sets of postulates are regarded merely as descriptions of mathematical structures. Their value consists in the fact that they are practical aids in the study of the mathematical structures that they describe...»

I've excerpted these paragraphs (emphasis in bold was mine) from:

EDWIN E. MOISE, Elementary geometry from an advanced standpoint. Addison-Wesley Publishing Company, Inc. Second Printing, March 1964, USA, pp. 382-383.

Incidentally, as I was browsing through some of the past volumes of The American Mathematical Monthly the other day, I found on page 776 of the eighth issue of Vol. 99 of that periodical a letter from an Alberto Guzmán (Dept. of Mathematics, City College of CUNY) to the Monthly Editors wherein Mr. Guzmán mentions that it was Alvin Hausner the one who called his attention to the fact that the change in viewpoint, “from accepting axioms as obvious truths to stipulating them as working assumptions”, was reflected in the Declaration of Independence and the Gettysburg Address. Mr. Guzmán wrote that letter because, in the first issue of the said volume of the Monthly, there appeared an article by Abe Shenitzer that touched upon the nineteenth-century change of standpoint in question and it, presumably, refreshed his memory on what Hausner had told him about the matter once. It has to be noted, however, that in the missive there was no mention whatsoever to either Lipman Bers or the paragraphs by Edwin Moise showcased above: the corollary being that even the Monthly Editors nod off sometimes.

The aforecited excerpts are also interesting because it is known that Lincoln was at some point in his life an avid reader of Euclid. In point of fact, some of his phrases—such as "dedicated to the proposition" in the Gettysburg Address—are attributed to his reading of Euclid. In addition, Lincoln is said to have spoken once thus2:

"... In the course of my law-reading I constantly came upon the word demonstrate. I thought, at first, that I understood its meaning, but soon became satisfied that I did not. I said to myself, 'What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof?' I consulted Webster's Dictionary. That told of 'certain proof,' 'proof beyond the possibility of doubt;' but I could form no idea what sort of proof that was. I thought a great many things were proved beyond a possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood 'demonstration' to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man. At last I said, 'Lincoln, you can never make a lawyer if you do not understand what demonstrate means;' and I left my situation in Springfield, went home to my father's house, and staid there till I could give any proposition in the six books of Euclid at sight. I then found out what 'demonstrate' means, and went back to my law studies."

The following comments by Salomon Bochner in “The Role of Mathematics in the Rise of Science” (Princeton University Press, 4th printing, Princeton NJ, USA, 1981, p. 37.) provide us with additional references on Lincoln's interest in Euclidean geometry:

“... Abraham Lincoln, in his campaign biography of 1860, written by himself and published under the name of John L. Scripps of the Chicago Press and Tribune, ventured to assert about himself that 'he studied and nearly mastered the six [sic] books of Euclid since he was a member of Congress.' (The Collected Works of Abraham Lincoln, The Abraham Lincoln Association, Springfield, Illinois (Rutgers University Press, 1953), IV, 62.) Lincoln's assertion that he had 'nearly mastered' these books was one of the boldest and blandest campaigns statements in the annals of the American presidential elections, and folkloristic embellishments of this assertion were even less restrained. (See Herndon's Life of Lincoln (The World Publishing Company, 1949); Carl Sandburg, Abraham Lincoln, The Prairie Years (Harcourt, Brace & Co., 1926), I, 423-424; Emanuel Hertz, Lincoln Talks (Viking, 1936), p. 18.) It is worth reflecting on the fact that in the America of 1860 a consummate grassroots politician of the then Mid-Western Frontier should have thought that adding to a mixture of log cabin and rail-splitting a six books worth of Euclid would make the mixture more palatable to an electorate across the country.”

Last but not least, I would like to add that Lincoln's devotion to Euclid was exploited in a scene of Steven Spielberg's 2012 movie on the Great Emancipator. As the Hindu mathematician Bhāskara would say (or so the legend has it), BEHOLD! 3

P.S. Please, feel free to enter below any observation, suggestion, criticism, etc. you may have for the owner/writer of this blog...

1 Jennnifer Schuessler (July 3, 2014). If only Thomas Jefferson could settle the issue (A period is questioned in the Declaration of Independence). The New York Times, p. A1.

2 Rev. J. P. Gulliver (September 4, 1864). Mr. Lincoln's early life: How he educated himself. The New York Times.

3 Be warned, though, that the short speech which Spielrock's Lincoln speaks in this scene is inaccurate in two or three respects.

viernes, 27 de marzo de 2015

Los problemas de matemáticas y mis problemas*

Ayer por la tarde, apenas acababa de sentarme a estudiar cuando llegó Licha mi hermana y me dijo:

—Toño, me dejaron un problema y no puedo resolverlo. Ayúdame, ¿quieres?

Así por encimita le eché un vistazo al problema y pensé en el compromiso tan grande que me echaba si no podía resolverlo porque perdería inmediatamente mi autoridad. Por eso le dije a Licha:

—Mira, ahorita no puedo ayudarte porque tengo mucho que estudiar. Vete a jugar un rato y cuando vuelvas te ayudaré con mucho gusto—así, pensé, mientras ella juega yo resuelvo el problema y luego se lo explico.

En cuanto Licha salió cogí su libreta y leí:

"Un niño y una niña fueron al bosque a buscar nueces. Recogieron 120 en total. La niña recogió la mitad de las que recogió el niño. ¿Cuántas nueces tenía el niño y cuántas la niña?"

Cuando terminé de leerlo hasta me dio risa: ¡uy, qué problemas les ponen en tercero!—pensé—. ¡Pero sí está todo clarísimo! Hay que dividir 120 entre 2 y resultarán 60. Luego, la niña recogió 60 nueces. Ahora hay que averiguar cuántas recogió el niño; de 120 me quitan 60, quedan otras 60. A ver, a ver, ¿cómo está esto? Así resulta que los dos recogieron la misma cantidad de nueces, pero el problema dice que la niña recogió la mitad de las que recogió el niño. ¡Ah! Entonces hay que dividir 60 entre 2 y tendremos 30. Luego, el niño recogió 60 nueces y la niña 30. Pero 60 y 30 son 90 y el problema dice que entre los dos recogieron 120 nueces.

—¡Pero que ocurrencia poner en tercer año un problema que no se puede resolver ni en cuarto!—pensé—. Eso es una injusticia...—la verdad era que sentía vergüenza de no poder resolverlo, pues Licha diría: "¿Ves? Estás en cuarto año y no puedes resolver un problema de tercero". Tenía que resolverlo a como diera lugar. Me puse a pensar de nuevo, pero no se me ocurrían otras soluciones. ¡Ya me había hecho bolas! Bueno, eran 120 nueces en total, y había que dividirlas de manera que el niño tuviera dos veces más que la niña. Desesperado, dibujé un nogal en el cuaderno, al pie del nogal una niña y un niño, y en el árbol 120 bolitas, que eran las nueces. Pero hasta ahí llegaba. Después, me puse yo a recoger nueces, es decir, a borrarlas del árbol y dárselas a los niños, dibujándoselas encima de la cabeza. Luego se me ocurrió que se las habían guardado en los bolsillos.

El niño tenía dos bolsillos en el pantalón y la niña sólo uno en su delantal. Entonces pensé que por eso la niña había recogido menos nueces que su hermano.

Estaba sentado, mirándolos: él tenía dos bolsillos, ella sólo uno. Y la cabeza empezó a despejárseme. Borré las nueces de encima de sus cabezas y dibujé de nuevo los bolsillos, pero esta vez eran unos bolsillos muy abultados, como si estuvieran llenos de nueces. Ahora las 120 nueces estaban dentro de los tres bolsillos. Entonces vi todo claro. ¡Cómo no se me había ocurrido antes! ¡Las 120 nueces había que dividirlas en tres partes! La niña toma una parte y el niño las partes restantes, es decir, dos veces más que la niña. Dividí rápidamente 120 entre 3 y resultó 40, las que tenía la niña. Y como el niño tenía el doble que ella, resultó que 40 más 40 daba 80. Luego sumé 80 y 40 y ¡eran las 120 nueces completitas!

Poco después regresó Licha e inmediatamente me puse a explicarle el problema. Le dibujé las nueces, los niños y sus bolsillos abultados.

—¡Qué bien explicas tú los problemas, Toño! Yo sola nunca habría sabido cómo hacerlo.

—Éste es un problema retefácil. Cuando te pongan uno más difícil me lo dices y yo te lo explico en un momento.

Entonces como que me envolvió una cosa muy bonita, como que me sentí muy importante de ver que yo podía ayudar a mi hermana a resolver sus problemas de matemáticas.

* Un cuento del escritor soviético Nikolái Nosov—adaptado al español por Armida de la Vara... Esta adaptación ha sido retomada del libro Español (Ejercicios y Lecturas) - Cuarto Grado, el cual estuvo vigente en México desde algún momento en los años 80 y hasta mediados de los años 90 (aproximadamente).

martes, 5 de agosto de 2014

Un cuento más: Cero en Geometría de Fredric Brown

Dicho cuento lo leí en un libro de texto hace muchos años. Recientemente me surgió el deseo de volver a leerlo pero como no recordaba ni los datos del libro en el cual lo había leído en aquella ocasión ni el autor del cuento el reencontrarlo fue todo un reto para la memoria... El planteamiento del cuento es clásico: al intentar salir avante de cierta problemática en su vida, el personaje principal decide invocar al diablo. Lo novedoso en el tratamiento de Brown es la extensión de su relato y el hilarante desenlace que le prepara al lector. La idoneidad del título del cuento —en español— será más que aparente al cabo de una primera lectura.

Les compartiré en esta entrada una transcripción del cuento basada en la que encontré en este sitio. No obstante es preciso mencionar que, atendiendo a la versión original del cuento, hice un par de correcciones a la adaptación en español que aparece en el enlace: donde en aquél sitio se encontraba la palabra pentágono (resp. hexágono) aparecerá ahora la palabra pentagrama (resp. hexagrama). Antes de ir al cuento en sí, agregaré algunos comentarios sobre los términos en pugna para el beneficio de los lectores más ocasionales de la bitácora.

Un pentágono es un polígono de cinco lados y con pentagrama nos estaremos refiriendo a lo largo de este post a la estrella de cinco puntas:

Por su parte, un hexágono es un polígono de seis lados y un hexagrama puede encontrarse, por ejemplo, en la bandera de Israel (de hecho, en el contexto judío la denominación más frecuente para la estrella de seis puntas es estrella de David):

Ahora sí, sin más dilaciones, presento a continuación el cuento al que se ha dedicado la entrada de este día:


Henry miró el reloj. Dos de la madrugada. Cerró el libro con desesperación. Seguramente que sería reprobado en el examen del día siguiente. Entre más estudiaba geometría, menos le entendía. Las matemáticas se le habían dificultado siempre pero la geometría le estaba resultando sencillamente imposible de aprender.

Lo peor era que no podía darse el lujo de reprobar la materia pues en sus primeros dos años en el colegio había reprobado ya otras tres y, de acuerdo con las estrictas reglas de su escuela, si ese año reprobaba una sola materia más sería eliminado automáticamente de los registros correspondientes de control escolar. Por otra parte, él certificado de compleción del colegio era indispensable para poder ingresar a la carrera que tenía contemplado estudiar. Sólo un milagro podría salvarlo.

Se levantó. ¿Un milagro? ¿Y por qué no? Siempre se había interesado en la magia. Tenía libros. Había encontrado instrucciones sencillísimas para llamar al diablo y someterlo a su voluntad. Nunca había hecho la prueba. Era el momento: ahora o nunca.

Sacó del estante el mejor libro sobre magia negra. Era fácil. Algunas fórmulas. Ponerse al abrigo en un pentagrama. El diablo llega. No puede nada contra uno y se obtiene lo que se quiera...

Movió los muebles hacia la pared, dejando el suelo limpio. Después dibujó sobre el piso, con un gis, el pentagrama protector. Procedió entonces a pronunciar las palabras cabalísticas. El diablo era horrible de verdad, pero Henry hizo acopio de valor y se dispuso a dictar su voluntad.

-- ―Siempre he tenido cero en geometría―empezó.

-- ―A quien se lo dices―contestó el diablo en un tono de burla.

Acto seguido, el diablo saltó las líneas del hexagrama que el muy idiota de Henry había dibujado, en lugar de un pentagrama, para devorarlo.