2-3 mai 2023 Chambéry - Le Bourget-du-Lac (France)

Exposés

Orateurs/Animateurs :        - Gokhan SOYDAN (Bursa Uludag University, Turkey),
                                             - Jean-Louis VERGER-GAUGRY (CNRS, University Savoie Mont-Blanc).
 

 

Partie I: Exposés 1, 2, 3   par  Gokhan SOYDAN,
Partie II: Exposé 4             par Jean-Louis VERGER-GAUGRY.

 (Nota : les résumés ci-dessous ne font pas l'objet d'une compilation LaTeX et certains symboles comme \neq, \equiv,... sont utilisés comme dans un source de fichier .tex .

nous espérons que le lecteur comprendra la mathématique, et l'encourageons à se reporter aux articles eux-mêmes)

 

Exposé No 1:    On the solutions of a class of generalized Fermat equations of signature (2, 2n, 3)
                            (Tuesday May 2 - 9:15-11:15 + 14:00-15:00 )

In this lecture, we first consider the Diophantine equation 7x2 + y2n = 4z3. We determine all
solutions to this equation for n = 2, 3, 4 and 5. We formulate a Kraus type criterion for showing that
the Diophantine equation 7x2 + y2p = 4z3 has no non-trivial proper integer solutions for speci c
primes p > 7. We computationally verify the criterion for all primes 7 < p < 109, p \neq 13. We use the
symplectic method and quadratic reciprocity to show that the Diophantine equation 7x2+y2p = 4z3
has no non-trivial proper solutions for a positive proportion of primes p.
Secondly, we consider the Diophantine equation x2 + 7y2n = 4z3, determining all families of
solutions for n = 2 and 3, as well as giving a (mostly) conjectural description of the solutions for
n = 4 and primes n  \geq 5.

This Lecture is based on [2, 3].

 


Exposé No 2:    On some Diophantine equations with power sums.
                            (Tuesday May 2 - 15:00-16:00 + Wednesday May 3 - 9:00-11:00)

Let k, l \geq  2 be fixed integers. In this lecture, we fi rst consider the Diophantine equation


                              (x + 1)k + (x + 2)k + ...+ (l x)k = yn,       x, y \in  Z, n \geq 2                               (0.2)

 Firstly, we prove that all solutions of the equation (0.2) in integers x, y, n with x, y \geq 1, n geq 2 satisfy
n < C1 where C1 = C1(l; k) is an e ffectively computable constant. Secondly, we prove that all
solutions of this equation in integers x, y, n with x, y \geq 1, n \geq 2 and k \neq 3 satisfy max{x, y, n} < C2
where C2 is an e ffectively computable constant depending only on k and l.
 
Next, we consider the Diophantine equation

                              (x - d)2 + x2 + (x + d)2 = yn                                                                                   (0.3)
 
Firstly, we give an explicit formula for all positive integer solutions of the equation (0.3) when n is
an odd prime and d = pr, p > 3 a prime. Secondly, under the assumption of our fi rst result, we
prove that the equation (0.3) has at most one solution (x, y). Thirdly, for a general d, we prove the
following two results: (i) if every odd prime divisor q of d satis es q \not\equiv  \pm 1 (mod 2n), then (0.3) has
only the solution (x, y, d, n) = (21, 11, 2, 3). (ii) if n > 228000 and d > 8 \sqrt{2}, then all solutions (x, y) of (0.3)
satisfy yn < 23/2 d3.
This lecture is based on [1], [4] and [6].
 
 
 
Exposé No 3:    On the solutions of some generalized Lebesgue-Ramanujan-Nagell equations
                             (Wednesday May 3 - 14:00-16:00)
 
In this lecture, we fi rst introduce generalized Lebesgue-Ramanujan-Nagell equations. Then, using
the modular approach (S. Siksek, [11]) , we show that if k \equiv 0 (mod 4), 30 < k < 724 and 2k - 1 is an odd prime
power, then under the GRH (generalized Riemann hypothesis), the generalized Ramanujan-Nagell
equation of the form x2+(2k-1)y = kz has only one positive integer solution (x, y, z) = (k-1, 1, 2).
The above results solve some dicult cases of Terai's conecture concerning this equation.
This Lecture is based on [5].
 
 

Exposé No 4:    Algebraic equations from a class of integer polynomials having lacunarity conditions
                             (Tuesday May 2 - 11:15-12:30 + Wednesday May 3 - 11:00- 12:30)
 
In this lecture we consider the class of integer polynomials :

C := { -1 + x + xn + xm1 + xm2 + ... + xms : n \geq  3,
                                                       m1 - n \geq n - 1,  mq - mq-1 \geq  n - 1,   for 2  \leq  q  \leq s }

Mellin's approach provides exact expressions of all the roots, for any n \geq 3 and any
s \geq 0. The case s = 0 corresponds to the trinomials. We show that it is not sufficient
to classify the roots, those close to the unit circle, those off the unit circle. We develop
the method of the asymptotic expansions (called ``Poincare") for the solutions of
-1 + x + xn = 0, for n \geq 3, and extend it to any polynomial of C. Those off the
unit circle form lenticuli. The set of lenticular roots can be completed and forms a
curve of solutions, image of a neighbourhood of 1 in (0; 1). To prove this, we show
that any polynomial P of C is uniquely associated to a Renyi numeration dynamical
system ([0, 1], Tb ) where Tb is the beta -transformation and 1/b is the unique zero of P
in (0, 1). The factorization of any P \in C is deduced from Ljunggren's method and
Kronecker's average value theorem. A Conjecture on the non-existence of reciprocal
non-cyclotomic components is formulated. For the reciprocal algebraic integers b > 1
very close to 1, using Kala-Vavra's theorem, we show an accumulation of conjugates of
on the completed lenticular curve. A (Dobrowolski-type) minoration of the Mahler
measure M( b) is deduced.
  This Lecture is based on [7-10].
 
 
References

[1] D. Bartoli and G. Soydan, The Diophantine equation (x+1)k +(x+2)k +...+(lx)k = yn
revisited, Publ. Math. Debrecen 96/1-2 (2020), 111--120.
[2] K. Chalupka, A. Dabrowski and G. Soydan, On a class of generalized Fermat equations
of signature (2, 2n, 3), J. Number Theory 34 (2022), 154--178.
[3] K. Chalupka, A. Dabrowski and G. Soydan, On a class of generalized Fermat equations
of signature (2, 2n, 3), II, submitted.
[4] M.-H. Le and G. Soydan,On the power values of the sum of three squares in arithmetic progression, Math. Commun.,
27 (2022), 137--150.
[5] E. K. Mutlu, M. H. Le and G. Soydan, A modular approach to the generalized Lebesgue-
Ramanujan-Nagell equation, Indagationes Mathematicae 33 (2022), 992--1000.
[6] G. Soydan, On the Diophantine equation (x + 1)k + (x + 2)k + ... + (lx)k = yn, Publ. Math.
Debrecen 91 (2017), 369--382.

[7] J.-L. Verger-Gaugry, On the Conjecture of Lehmer, limit Mahler measure of trinomials and
asymptotic expansions, Uniform Distribution Theory J. 11 (2016), 79--139.
[8] D. Dutykh, J.-L. Verger-Gaugry, Alphabets, rewriting trails, periodic representation in algebraic
basis, Res. Number Theory 7:64 (2021).
[9] J.-L. Verger-Gaugry, A proof of the Conjecture of Lehmer
math NT> arXiv:1911.10590 (29 Oct 2021), 114 pages.
[10] D. Dutykh, J.-L. Verger-Gaugry, On a class of lacunary almost-Newman polynomials modulo
p and density theorems, Unif. Distrib. Theory 17, No 1 (2022), 29--54.
 
 
The modular approach:
 
[11] S. Siksek, The Modular Approach to Diophantine Equations, IHPNotes.
 

Recent:

(1) Elif Kizildere Mutlu, Maohua Le and G. Soydan, An elementary approach to the generalized
Ramanujan-Nagell equation, Indian Journal of Pure and Applied Mathematics, (2023), to appear.
(2) J.-L. Verger-Gaugry, A Dobrowolski-type inequality for the poles of the dynamical zeta function
of the b-shift, submitted (2022).
https://hal.science/hal-03754732v1
(3) J.-L. Verger-Gaugry, An universal minoration of the Mahler measure of real reciprocal algebraic
integers, (2022).
https://hal.science/hal-03754750v1
(4) J.-L. Verger-Gaugry, A Panorama on the Minoration of the Mahler Measure: from the Problem
of Lehmer to its Reformulations in Topology and Geometry,
https://hal.science/hal-03148129v1
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