(Japanese / 日本語)
(Japanese / 日本語)
Speaker: István Pink (University of Debrecen)
Date: 2021 April 17th (Sat) from 16:00 JST
Title: Number of solutions to a special type of unit equations in two unknowns
Abstract: For any fixed coprime positive integers a,b and c with min{a,b,c}>1,
we prove that the equation a^x+b^y=c^z has at most two solutions in positive integers x,y and z,
except for one specific case which exactly gives three such solutions.
Our result is essentially sharp in the sense that there are infinitely many examples
allowing the equation to have two solutions in positive integers.
From the viewpoint of a well-known generalization of Fermat's equation,
it is also regarded as a 3-variable generalization of the celebrated theorem of Bennett (2001)
which asserts that Pillai's type equation a^x-b^y=c has at most two solutions
in positive integers x and y for any fixed positive integers a,b and c with min{a,b}>1.
In this talk we give a brief summary of corresponding earlier results
and present the main improvements leading to this definitive result.
This is a joint work with Takafumi Miyazaki (Gunma University).
Speaker: Yasutsugu Fujita（Nihon University）
Date: 2021 May 22nd (Sat) from 16:00 JST
Title: Integral points and generators for the Mordell-Weil groups of elliptic curves
Abstract: (English) abstract
Speaker: Wataru Takeda （Tokyo university of science）
Date: 2021 June 19th (Sat) from 16:00 JST
Title: Transcendence of the iterated exponential of algebraic numbers
Abstract: We study the transcendence of the limit h(A) of the sequence: A, A^A, A^{A^A},... In 2010, Sondow and Marques studied the case that A is rational numbers or algebraic numbers satisfying some special conditions. In this talk, we extend their results and give an asymptotic formula for the number of algebraic numbers A such that h(A) is algebraic. We also obtain the distribution of such algebraic numbers. In the end of this talk, we talk about our recent progress and future works. This is the joint work with Hirotaka Kobayashi(Nagoya University) and Kota Saito(Nagoya University).
Speaker: Shin-ichiro Seki （Aoyama Gakuin University）
Date: 2021 July 17th (Sat) from 16:00 JST
Title: On a generalization of the Green-Tao theorem to number fields
Abstract: The Green-Tao theorem which states that there exist arithmetic progressions of primes of arbitrary length is a milestone in additive number theory. Tao also proved a version of this theorem for the field of Gaussian numbers and conjectured that the Green-Tao theorem could be extended to the case of general number fields. On the other hand, his proof depends on the fact that the class number of the field of Gaussian numbers is 1 and the unit group of the ring of Gaussian integers is a finite group. Although he wrote that his proof is also likely to extend to other number fields if they are in the same situation, there are only ten such number fields. Recently, the speaker, in collaboration with Kai, Mimura, Munemasea, Yoshino, has succeeded in proving the Green-Tao theorem for general number fields without any conditions by constructing appropriate arguments. In the first half of the seminar, I will introduce various results including previous studies, and in the second half, I will explain the outline of the proof in an omnibus style with emphasis on each key point.
Speaker: Yuta Suzuki （Rikkyo University）
Date: 2021 September 25th (Sat) from 16:00 JST
Title: On even-odd amicable numbers
Abstract: A pair of positive integers (A,B) is called an amicable pair if the sum of all proper divisors of A equals B and the sum of all proper divisors of B equals A. It is one of the folklore conjectures that there is no amicable pair consisting of even and odd numbers. It is easy to see that if there is an even-odd amicable pair (A,B), then it should be of the form A=2^aM^2, B=N^2 with odd numbers M,N and a positive integer a up to permutation. Therefore, roughly speaking, even-odd amicable pairs are a "quadratic subclass" of the whole amicable pairs. In this talk, we extend the method of Pomerance to bound the number of amicable pairs to the even-odd amicable pairs and obtain an upper bound for the number of the even-odd amicable pairs which is naturally expected for the "quadratic subclass". This bound is an improvement of the former bound remarked by Pollack.
Speaker: Dong Han KIM （Dongguk University）
Date: 2021 October 23rd (Sat) from 16:00 JST
Title: On the repetition of Sturmian word and numbers with Sturmian expansions
Abstract: In this talk, we discuss repetition properties of Sturmian words in combinatorial ways. These combinatorial properties are applied to study irrationality exponents of numbers of Sturmian b-ary expansions and to construct a Cantor set without algebraic points except for the two end points. We then briefly survey the following results on the spectrum of irrationality exponent. We also consider the Lévy constant of numbers with Sturmian continued fraction expansions.
Speaker: Yann Bugeaud （University of Strasbourg）
Date: 2021 November 13th (Sat) from 16:00 JST
Title: Combinatorial structure of Sturmian words and continued fraction expansions of Sturmian numbers
Abstract: Let $\theta$ be an irrational real number in $(0, 1)$. It is well-known that the characteristic Sturmian word of slope $\theta$ is the limit of a sequence of finite words $(M_k)_{k \ge 0}$, with $M_k$ of length $q_k$ (the denominator of the $k$-th convergent to $\theta$) being a suitable concatenation of copies of $M_{k-1}$ and one copy of $M_{k-2}$. We extend this to any Sturmian word. Let $b \ge 2$ be an integer. As an application, we give the continued fraction expansion of any real number $\xi$ whose $b$-ary expansion is a Sturmian word ${\bf x}$ over the alphabet $\{0, b-1\}$. This extends a classical result of B\"ohmer (1927) who considered only the case where ${\bf x}$ is characteristic. Consequently, we obtain a formula for the irrationality exponent of $\xi$ in terms of the slope and the intercept of ${\bf x}$. This is a joint work with Michel Laurent.
Speaker: Kurosawa Takeshi (Tokyo University of Science)
Date: 2021 December 18th (Sat) 2021 from 16:00 JST
Title: The Hone-type continued fraction expansion and its irrationality exponent
Abstract: We give a generalization of the regular continued fraction expansions discovered by Hone and Varona. The continued fraction expansion we discuss is non-regular. In the first part of the talk, we introduce the expansion and move to the irrationality exponent of the continued fraction in the second part.
Speaker: Haruki Ide (Keio University)
Date: 2022 January 22nd (Sat) from 16:00 JST
Title: Algebraic independence of the values and the derivatives of a certain family of Lambert-type series
Abstract: (English) abstract
Speaker: Takumi Noda (Nihon University)
Date: 2022 February 19th (Sat) 2022 from 16:00JST
Title: Transformations and asymptotics for a class of Dirichlet-Hurwitz-Lerch Eisenstein series
Abstract: (English) abstract
Speaker: Stéphane Fischler (Universite Paris-Saclay)
Date: 19th March 2022 (Sat) from 16:00 JST
Title: Linear independence of odd zeta values using Siegel's lemma
Abstract: Conjecturally, 1 and all values of the Riemann zeta function at odd integers s ≥ 3 are linearly independent over the rationals (and these zeta values are, therefore, irrational). However, very few is known in this direction. Apéry proved in 1978 that ζ(3) is irrational; Ball-Rivoal proved in 2001 that for any ε >0, at least (1-ε) (log s) / (1+log 2) numbers among 1, ζ(3), ζ(5), ..., ζ(s) are linearly independent over the rationals, when s is odd and large enough in terms of ε. In this lecture we shall explain how this lower bound can be improved to 0.21 \sqrt{s/log s}. The strategy is to replace explicit constructions with the use of Siegel's lemma.