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The article attempts to demonstrate the rich history of one truly remarkable problem situated at the confluence of probability theory and theory of numbers - finding the probability of co-primality of two randomly selected natural numbers.…
The purpose of this memoir is to discuss two very interesting properties of integer sequences. One is the law of apparition and the other is the law of repetition. Both have been extensively studied by mathematicians such as Ward, Lucas,…
The main purpose of this paper is to propose some interesting number theory problems related to the Legendre's symbol and the two-term exponential sums.
The main purpose of this survey is to introduce an inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic…
Tensors are ubiquitous in statistics and data analysis. The central object that links data science to tensor theory and algebra is that of a model with latent variables. We provide an overview of tensor theory, with a particular emphasis on…
This article is the rejoinder for the paper "Probabilistic Integration: A Role in Statistical Computation?" to appear in Statistical Science with discussion. We would first like to thank the reviewers and many of our colleagues who helped…
Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic…
One challenge (or opportunity!) that many instructors face is how varied the backgrounds, abilities, and interests of students are. In order to simultaneously instill confidence in those with weaker preparations and still challenge those…
This article seeks to encourage a mathematical dialog regarding a possible solution to Beals Conjecture. It breaks down one of the worlds most difficult math problems into laymans terms and encourages people to question some of the most…
This is a pedagogical article cited in the foregoing research note, quant-ph/9911050
I revisit an automated proof of Andrews' pentagonal number theorem found by Riese. I uncover a simple polynomial identity hidden behind his proof. I explain how to use this identity to prove Andrews' result along with a variety of new…
The features of a logically sound approach to a theory of statistical reasoning are discussed. A particular approach that satisfies these criteria is reviewed. This is seen to involve selection of a model, model checking, elicitation of a…
This paper has been withdrawn. With the advancement of statistical theory and computing power, data sets are providing a greater amount of insight into the problems of today. Statisticians have an ever increasing number of tools to attack…
Comparative prime number theory is the study of the {\em{discrepancies}} of distributions when we compare the number of primes in different residue classes. This work presents a list of the problems being investigated in comparative prime…
The rather unintuitive nature of quantum theory has led numerous people to develop sets of (physically motivated) principles that can be used to derive quantum mechanics from the ground up, in order to better understand where the structure…
Research at the intersection of machine learning and the social sciences has provided critical new insights into social behavior. At the same time, a variety of critiques have been raised ranging from technical issues with the data used and…
The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be…
Translated from the Latin original, "De numeris amicabilibus" (1747). E100 in the Enestroem index. Euler starts by saying that with the success of mathematical analysis, number theory has been neglected. He argues that number theory is…
We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.
In this paper we define a function $L$ which will allow us to (separately or simultaneously) generalize many theorems from Number Theory obtained by Wilson, Fermat, Euler, Gauss, Lagrange, Leibniz, Moser, and Sierpinski.