This Workshop of Number Theory (May 2-3 2023) is devoted to Diophantine Equations and Algebraic Equations. It is integrated in the Erasmus Programme between the University Savoie Mont Blanc (USMB) and the University Bursa Uludag (Turkey). The Department of Mathematics (LAMA) of USMB is organizing this event.

Participation is only on-line. Registration is compulsory and free. Once registered, a ZOOM link will be sent to the participants.

In the Workshop 4 Lectures will be given, by two speakers: G. Soydan, J.-L. Verger-Gaugry.

Deadline for registration (recommended) : Friday April 28  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 = yⁿ,             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 xⁱ, as a function of the degree n and the coefficient vector (a₀, a₁,... , 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 + xⁿ, 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+xⁿ 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].