Related papers: Baker's Explicit abc-Conjecture and applications
Approximate Bayesian computation (ABC) has advanced in two decades from a seminal idea to a practically applicable inference tool for simulator-based statistical models, which are becoming increasingly popular in many research domains. The…
We extend a conjecture of Kimberley-Robertson on the abelianizations of certain square complex groups.
We show that the abc Conjecture implies the Weak Diversity Conjecture of Bilu and Luca.
Recent developments of Baxter algebras have lead to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first kind and the second kind, partitions and multinomial…
We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with $a+b=c$ and $c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby…
We establish the exact overlaps conjecture for iterated functions systems on the real line with algebraic contractions and arbitrary translations.
We consider a variant of the ABC Conjecture, attempting to count the number of solutions to $A+B+C=0$, in relatively prime integers $A,B,C$ each of absolute value less than $N$ with $r(A)<|A|^a, r(B)<|B|^b, r(C)<|C|^c.$ The ABC Conjecture…
In this paper, authors study the convexity and concavity properties of real-valued function with respect to the classical means, and prove a conjecture posed by Bruce Ebanks in \cite{e}.
We present an astonishingly simple and elegant proof of the celebrated Basel problem.
The well-known $abc$-conjecture concerns triples $(a,b,c)$ of non-zero integers that are coprime and satisfy ${a+b+c=0}$. The strong $n$-conjecture is a generalisation to $n$ summands where integer solutions of the equation ${a_1 + \ldots +…
The Hodge conjecture is a major open problem in complex algebraic geometry. In this survey, we discuss the main cases where the conjecture is known, and also explain an approach by Griffiths-Green to solve the problem.
We survey Vojta's higher-dimensional generalizations of the $abc$ conjecture and Szpiro's conjecture as well as recent developments that apply them to various problems in arithmetic dynamics. In particular, the "$abcd$ conjecture" implies a…
We establish an equivalent condition to the validity of the Collatz conjecture, using elementary methods. We derive some conclusions and show several examples of our results. We also offer a variety of exercises, problems and conjectures.
As a first application of a very old theorem, known as Herschel's theorem, we provide direct elementary proofs of several explicit expressions for some numbers and polynomials that are known in combinatorics. The second application deals…
In this note, we present a conjecture on intersections of set families, and a rephrasing of the conjecture in terms of principal downsets of Boolean lattices. The conjecture informally states that, whenever we can express the measure of a…
This paper is an extension program of the notion of circle of partition developed in our first paper \cite{CoP}. As an application we prove the Erd\H{o}s-Tur\'{a}n additive base conjecture.
Using Balmer--Favi's generalized idempotents, we establish the telescope conjecture for many algebraic stacks. Along the way, we classify the thick tensor ideals of perfect complexes of stacks.
In a recent paper, Merca posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalized $m$-gonal numbers. Extending Merca's work, we complete the proof of these…
We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.
A simple expression is derived for the terms in the Baker-Campbell-Hausdorff series. One formulation of the result involves a finite number of operations with matrices of rational numbers. Generalizations are discussed.