Cet Atelier de Théorie des Nombres (2-3 mai 2023) est consacré aux équations Diophantiennes et aux équations algébriques. Il est uniquement en visio et s'inscrit dans le cadre Erasmus entre l'Université Savoie Mont Blanc (USMB) et l'Université Bursa Uludag (Turquie). Le Département de Mathématiques (LAMA) de l'USMB est organisateur de cet évènement.

L'inscription est obligatoire et gratuite pour pouvoir participer. A chaque mèl de participant inscrit sera envoyé le lien ZOOM pour participer pendant les deux jours de l'atelier.

L'Atelier comporte 4 exposés et deux orateurs : G. Soydan, J.-L. Verger-Gaugry.

Date limite pour s'inscrire (recommandée) : vendredi 28 avril 2023.

**INTRODUCTION**

The main purpose of this workshop is to bring together PhD students and researchers who

are interested in algebraic and/or Diophantine equations.

In Part I, we first show how to apply the modular method [11] to a class of generalized Fermat equations and certain generalized Lebesgue- Ramanujan-Nagell type equations. Secondly, introducing some known results on Schaffer's Conjecture we will show that how to use Bernoulli numbers/polynomials for solving some Diophantine

equations with power sums.

In Part II we are interested in the set of zeroes of univariate lacunary

polynomials in a special class of integer polynomials having coefficients in {0, 1}, except the constant

term equal to -1, and a gappiness whose size is controlled a minima by an unique integer. We establish

the link between this class of integer polynomials and a dense subset of non-reciprocal algebraic

integers b > 1 close to 1, by the Renyi numeration dynamical system and the b-transformation. We

show that the completion of this class of zeroes contains an analytic curve which contains Galois

conjugates of the bs > 1 which are reciprocal algebraic integers close to 1. Some consequences on

the minoration of the Mahler measure M(b) of such bs are evoked.

Part I: A great deal of number theory arises from the discussion of the integer or rational

solutions of a polynomial equation with integer coefficients. Such equations are called Diophantine

equations. In 1637, Pierre de Fermat conjectured that the equation x^{p} + y^{p} = z^{p} has no solutions

in non-zero integers x, y, z for p \geq 3. This is known as Pierre de Fermat's Last Theorem. In 1995, the most

important progress in the field of the Diophantine equations has been with Andrew Wiles' proof of

Pierre de Fermat's Last Theorem. His proof is based on deep results about Galois representations associated

to elliptic curves and modular forms. The method of using such results to deal with Diophantine

problems, is called the modular approach.

In the same century, Jakob Bernoulli (1655-1705) introduced the Bernoulli numbers in connection

to the study of the sums of powers of consecutive integers 1^{k} + 2^{k} +...+ n^{k}. The study of the

polynomial Diophantine equation in the form of

1^{k} + 2^{k} + ... + x^{k} = y^{n}, x, y \in** Z**^{+}, n \geq 2 (0.1)

has been going on for more than a hundred years. In 1956, Juan Schaffer showed, for k \geq 1 and

n \geq 2, that (0.1) possesses at most finitely many solutions in positive integers x and y, unless

(k, n) \in {(1, 2), (3, 2), (3, 4), (5, 2)}, where, in each case, there are infinitely many such solutions.

Sch"affer's conjectured that (0.1) has the unique non-trivial (i.e. (x, y) \neq (1, 1)) solution, namely

(k, n, x, y) = (2, 2, 24, 70). The correctness of this conjecture has been proved for some cases. But,

it has not been proved completely yet.

Part II: The question of finding exact expressions for the roots of integer polynomials, say

S_{{i=0}}^{{n}} a_{i} x^{i}, as a function of the degree n and the coefficient vector (a_{0}, a_{1},... , a_{n}), and understanding

the role of lacunarity on the factorization, has received a lot of attention. Let us mention the

Renaissance mathematicians at Bologna, Fontana, Tschirnhaus, Leibniz, Lagrange, Abel, Galois,

Liouville, for the method of radical expressions (multiplication, addition, subtraction, division, extraction

of roots, only permitted on the coefficients). Galois theory says that the case n = 5, for the

degree, is a critical case above which other techniques than radical expressions should be applied

for solving the equations. Mellin (1915) had overcome this difficulty by solving general algebraic

equations by suitable Cauchy integration in the complex plane using hypergeometric functions and

the Gamma-function (G. Belardinelli, Fonctions hypergéométriques de plusieurs variables et résolution analytique

des équations algéebriques générales, Mem. Sci. Math. Fasc. 145, Gauthiers-Villars, Paris, 1960). Later, the role of lacunarity on the factorization was investigated by Schinzel, Dobrowolski, in terms of cyclotomic, reciprocal noncyclotomic and nonreciprocal components, and generalized to several variables.

In this talk attention will be focused on a class of almost-Newman integer polynomials which

are sums of a trinomial -1 + x + x^{n}, n \geq 3, and a perturbation Newman polynomial having

distanciation between successive monomials greater than or equal to n-1. Though Mellin's approach

be applicable, for all the roots, the method of asymptotic expansions of the roots of the trinomials

-1+x+x^{n} which is developed (in [7]) is more powerful to give access to more precise formulations

and properties of a subcategory of roots, called lenticular. Consequences are evoked in [7] [8] [9]

[10].