PROGRAM 
Room 1 of the IFM 
9:00—9:30 Carl Pomerance (Dartmouth College, USA) Title: Square Values of Euler’s function. Abstract: 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).
9:40—10:10 Pedro Berrizbeitia (University of Simon Bolivar, Venezuela) Title: Sequences of Integers with Fast Primality Test. Abstract: The fastest known primality test are the LucasLehmer 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 pth Mersenne number M_{p}=2^{p}1 by computing p2≈log(M_{p})2 modular squares (squares mod M_{p}). The second one, discovered by Pépin on the following year, determines the primality of the nth Fermat number F_{n} computing log(F_{n})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.
10:20—10:50 Pantelimon Stanica (Naval Postgraduate School, USA) Title: Attacks on RSA and its Variations. Abstract: In this talk we will survey some known and perhaps less known attacks on RSA and some variations of it.
11:00—11:30 Amanda Montejano (National Autonomous University of Mexico, Mexico) Title: The Use of Additive tools in Solving Arithmetic AntiRamsey Problems. Abstract: The study of the existence of rainbow structures in colored universes falls into the antiRamsey 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 rainbowfree colorings. Beyond of studying density conditions we want to characterize the structure of rainbowfree 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
12:00—12:30 Yuri F. Bilu (University of Bordeaux, France) Title: Drawing Curves on Checked Paper. Abstract: 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).
12:40—13:10 Amalia Pizarro (University of Valparaiso, Chile) Title: Explicit Formulas and Lfunctions. Abstract: Explicit formulas relate sums over the primes and complex zeros of Lfunctions. In this talk, we survey some applications of this method in the study of the properties of Lfunction. In particular, we are interested in Lfunctions attached to an holomorphic modular form and Artin Lfunctions.
13:20—13:50 Gabriel Villa Salvador (Cinvestav, Mexico) Title: Radical Extensions and the Carlitz Module. Abstract: A finite extension L/K is called radical if L is generated over K by torsion elements of the Zmodule L*/K*. The study of this kind of extensions involves several concepts and special families of extensions: cogalois extensions, Kneser extensions, Gcogalois extensions, hereditary Kneser extensions, ΓClifford 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 Zmodule structure and also the action given by the Carlitzmodule. In this talk we present some results on radical extensions and Kummer extensions defined by the CarlitzHayes action instead of the Zaction. This is a joint work with Marco Antonio SanchezMirafuentes.
13:50 Closing Ceremony. 