Related papers: Baker's Explicit abc-Conjecture and applications
The author has recently presented two different expressions for lambda_d(p) of the monomer-dimer problem involving a power series in p, the first jointly with Shmuel Friedland. These two expressions are certainly equal, but this has not yet…
We prove that the frequency of abc equations c^n = a+b satisfying the strong abc - conjecture is phi(c^n)/2+o(phi(c^n)/2), for n going to infinity.
In this paper, we will give an extension of Mok's theorem on the generalized Frankel conjecture under the condition of the orthogonal bisectional curvature.
In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the…
In the paper we complete a case by case proof of Reeder's Conjecture started in our previous work, proving the conjecture for simple Lie algebras of type $D$ and for the exceptional cases.
The $abc$ conjecture states that there are only finitely many triples of coprime positive integers $(a,b,c)$ such that $a+b=c$ and $\operatorname{rad}(abc) < c^{1-\epsilon}$ for any $\epsilon > 0$. Using the optimized methods in a recent…
This document is due to appear as a chapter of the forthcoming Handbook of Approximate Bayesian Computation (ABC) edited by S. Sisson, Y. Fan, and M. Beaumont. Since the earliest work on ABC, it has been recognised that using summary…
We study the size of the exceptional set in the $abc$ conjecture, improving on and simplifying work of Browning, Lichtman and Ter\"av\"ainen.
The ABC conjecture implies many conjectures and theorems in number theory, including the celebrated Fermat's Last Theorem. Mason-Stothers Theorem is a function field analogue of the ABC conjecture that admits a much more elementary proof…
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that…
In this paper, we study the Br\"{u}ck conjecture \cite{Bruck-1996} by interpreting it through solutions of first-order partial differential equations in several complex variables. Our results show that the Br\"{u}ck conjecture…
We prove Ma\~n\'e's conjectures in the context of codimension one Aubry-Mather theory
Two prominent conjectures by Herbert J. Ryser have been falsely attributed to a somewhat obscure conference proceedings that he wrote in German. Here we provide a translation of that paper and try to correct the historical record at least…
We give a proof of a conjecture of A. Lacasse in his doctoral thesis which has applications in machine learning algorithms. The proof relies on some interesting binomial sums identities introduced by Abel (1839), and on their generalization…
We aim to prove a twisted version of the Osborne conjecture obtained by Hecht and Schmid in their 1983 Acta Mathematica paper. Bergeron and Clozel (2013) have considered a special case, and we generalize their method to our setting.
The $abc$ conjecture predicts a highly non trivial upper bound for the height of an algebraic point in terms of its discriminant and its intersection with a fixed divisor of the projective line counted without multiplicity. We describe the…
We present a conjecture about partitions, with a very elementary formulation.
In this paper, we define $X$-base Fibonacci-Wieferich prime which is a generalized Wieferich prime where $X$ is a finite set of algebraic numbers. We are going to show that there are infinitely many non-$X$-base Fibonacci-Wieferich primes…
The choice of the summary statistics used in Bayesian inference and in particular in ABC algorithms has bearings on the validation of the resulting inference. Those statistics are nonetheless customarily used in ABC algorithms without…
Approximate Bayesian Computation (ABC for short) is a family of computational techniques which offer an almost automated solution in situations where evaluation of the posterior likelihood is computationally prohibitive, or whenever…