Rectángulo redondeado: MORELIA NUMBER THEORY DAY
AUGUST 2, 2013


Room 1 of the IFM


Carl Pomerance (Dartmouth College, USA)

Title: Square Values of Euler’s function.


Euler’s function, ubiquitous in number theory, has been studied as an arithmetic function for a long time. We know it’s average to x fairly well, we know it’s distribution, and we know a lot about the frequency of integers (called totients) which are values of Euler’s function. This talk concerns square totients. It is perhaps surprising that up to x there are many more integers that Euler’s function maps to a square than there are squares themselves. However, are most squares totients? Surely the answer should be “no” and in this paper we prove this. Perhaps there is an easier path, but our proof is surprisingly difficult. (Joint work with Paul Pollack).



Pedro Berrizbeitia (University of Simon Bolivar, Venezuela)

Title: Sequences of Integers with Fast Primality Test.


The fastest known primality test are the Lucas-Lehmer Test for Mersenne numbers and the Pépin Test for Fermat numbers. The first one was discovered by Lucas in 1876. It allows to determine the primality of the p-th Mersenne number Mp=2p-1 by computing p-2log(Mp)-2 modular squares (squares mod Mp). The second one, discovered by Pépin on the following year, determines the primality of the n-th Fermat number Fn computing log(Fn)-1 modular squares. The problem that motivated this investigation was to find sequences of integers in which primality could be determined faster than that. In this talk we present two such sequences together with their corresponding test: the Cullen numbers and the Fermat numbers. These tests determine the primality of a prime n by computing log(n)-log(log(n))-C modular squares. We end by discussing the scope of the ideas involved. 



Pantelimon Stanica (Naval Postgraduate School, USA)

Title: Attacks on RSA and its Variations.


In this talk we will survey some known and perhaps less known attacks on RSA and some variations of it.



Amanda Montejano (National Autonomous University of Mexico, Mexico)

Title: The Use of Additive tools in Solving Arithmetic Anti-Ramsey Problems.


The study of the existence of rainbow structures in colored universes falls into the anti-Ramsey theory. Canonical versions of this theory prove the existence of either a monochromatic structure or a rainbow structure. Instead, in the most recent so called rainbow Ramsey theory, the existence of rainbow structures is guaranteed regardless of the existence of monochromatic structures. Arithmetic versions of this theory were initiated by Jungic, Fox, Mahdian, Nesetril and Radoicic studying the existence of rainbow arithmetic progressions in colorings of cyclic groups and of intervals of integers. In this setting, it happens most of the times that to ensure the existence of a rainbow structure the color classes have to satisfy some density conditions. Our particular interest is in describing colorings containing no rainbow structures. We called this colorings rainbow-free colorings. Beyond of studying density conditions we want to characterize the structure of rainbow-free colorings. In this talk we present how to use classical inverse theorems in additive number theory in order to obtain results following this philosophy.


11:30—12:00 Coffee Break



Yuri F. Bilu (University of Bordeaux, France)

Title: Drawing Curves on Checked Paper.


Imagine a sheet of checked paper, the one used in arithmetic classes in primary schools. Let us try to draw a curve on this sheet which would intersect as many "crossings" as possible. (A mathematician would say: let how many lattice points a compact curve can meet?) Of course, one can draw a rather "curvy" curve which would pass through every crossing. But the problem becomes interesting if the curve is assumed "not too curvy". For instance, in 1927 the Czech mathematician Jarnik proved that a *strictly convex* curve can pass at most O(N^{2/3}) crossings on an NxN checked sheet. I will prove the theorem of Jarnik using an argument suggested by the German mathematician Dörge (1926), who worked independently of Jarnik on related problems. I will then show how a slight modification of Dörge's argument leads to a wonderful theorem of Bombieri and Pila (1989) on counting lattice points on analytic curves. Finally, if time permits, I will briefly describe the recent development of these ideas in the domain of "unlikely intersections" (important works by Pila, Zannier and others). 

The lecture will require very little background. The first half does not use anything beyond the Mean Value Theorem. In the second part it would be worth knowing what a real analytic function is, but this would be really required only at the very end. 



Amalia Pizarro (University of Valparaiso, Chile)

Title: Explicit Formulas and L-functions.


Explicit formulas relate sums over the primes and complex zeros of L-functions. In this talk, we survey some applications of this method in the study of the properties of L-function. In particular, we are interested in L-functions attached to an holomorphic modular form and Artin L-functions.



Gabriel Villa Salvador (Cinvestav, Mexico)

Title: Radical Extensions and the Carlitz Module.


A finite extension L/K is called radical if L is generated over K by torsion elements of the Z-module L*/K*. The study of this kind of extensions involves several concepts and special families of extensions: cogalois extensions, Kneser extensions, G-cogalois extensions, hereditary Kneser extensions, Γ-Clifford extensions, abstract cogalois theory and so on. A special class of radical extensions is the class of Kummer extensions. In case that L and K are congruence function fields, we have the Z-module structure and also the action given by the Carlitz-module. In this talk we present some results on radical extensions and Kummer extensions defined by the Carlitz-Hayes action instead of the Z-action. This is a joint work with Marco Antonio Sanchez-Mirafuentes.


13:50 Closing Ceremony.