Related papers: Open Conjectures on Congruences
I review the classical theory of likelihood based inference and consider how it is being extended and developed for use in complex models and sampling schemes.
The possible tensor constructions of open string theories are analyzed from first principles. To this end the algebraic framework of open string field theory is clarified, including the role of the homotopy associative A_\infty algebra, the…
In this paper we derive congruences expressing Bell numbers and derangement numbers in terms of each other modulo any prime.
In this short survey article, we aim to provide an up to date information on the progress made towards Schurs exponent conjecture and related conjectures. We also mention the connection between Schurs exponent conjecture and Noether's…
Prediction is a complex notion, and different predictors (such as people, computer programs, and probabilistic theories) can pursue very different goals. In this paper I will review some popular kinds of prediction and argue that the theory…
We survey some results that provide different versions of classical results through different summability methods. Specifically, in order to adapt such classical results, we analyze which properties should satisfy the summability methods.…
This paper presents an offering of some of the myriad connections between Combinatorics and Probability, directed in particular toward combinatorialists. The choice of material was dictated by the author's own interests, tastes and…
This paper presents and discusses several methods for reasoning from inconsistent knowledge bases. A so-called argumentative-consequence relation taking into account the existence of consistent arguments in favor of a conclusion and the…
In this survey, we give a short overview of the recent progress on the multidimensional L2 conjecture. It can also serve as an introduction to the subject.
In this paper, we are interested in the interplay between integral ternary quadratic forms and class numbers. This is partially motivated by a question of Petersson.
In this note some philosophical thoughts and observations about mathematics are expressed, arranged as challenges to some common claims.
Mathematicians had little idea whether the easy-to-state union-closed conjecture was true or false even after $40$ years. However, last winter saw a surge of interest in the conjecture and its variants, initiated by the contribution of a…
The article provides a counterexample to a conjecture by Blocki-Zwonek.
Work in progress concerning alternative formalizations of arithmetic.
We report the results of our empirical investigations on the Bateman-Horn conjecture. This conjecture, in its commonly known form, produces rather large deviations when the polynomials involved are not monic. We propose a modified version…
In this paper, we will present several new congruences involving binomial coefficients under integer moduli, which are the continuation of the previous two work by Cai \textit{et al.} (2002, 2007).
The aim of this expository article is twofold. The first is to introduce several polynomials of one variable as well as two variables defined on the positive integers with values as congruent numbers. The second is to present connections…
Ellenberg proved that the abc conjecture would follow if this conjecture were known for sums $a+b=c$ such that $D\mid abc$ for some integer~$D$. Mochizuki proved a theorem with an opposite restriction, that the full abc conjecture would…
Given a random real quadratic field from $\{ \mathbb{Q}(\sqrt{p}\,) ~|~ p \text{ primes} \}$, the conjectural probability $\mathbb{P}(h=q)$ that it has class number $q$ is given for all positive odd integers $q$. Some related conjectures of…
Breakthrough discoveries and inventions involve unexpected combinations of contents including problems, methods, and natural entities, and also diverse contexts such as journals, subfields, and conferences. Drawing on data from tens of…