A Pascal-Type Triangle Characterizing Twin Primes

Large primes in generalized Pascal triangles

In this paper, after presenting the results of the generalization of Pascal triangle (using powers of base numbers), we examine some properties of the 112-based triangle, most of all regarding to prime numbers. Additionally, an effective implementation of ECPP method is presented which enables Magma computer algebra system to prove the primality of numbers with more than 1000 decimal digits. 1 ...

Erdos and the Twin Prime Conjecture: Elementary Approaches to Characterizing the Differences of Primes

The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis

We define the Pascal triangle of a discrete (gray scale) image as a pyramidal arrangement of complex-valued moments and we explore its geometric significance. In particular, we show that the entries of row k of this triangle correspond to the Fourier series coefficients of the moment of order k of the Radon transform of the image. Group actions on the plane can be naturally prolonged onto the e...

Research Statement Dominic Klyve

My research concerns explicit inequalities in elementary number theory. Specifically, I am interested in the distribution of twin primes, along with theoretical and computational techniques for putting explicit bounds on classes numbers such as twin primes. One important method I use to examine the density of twins is to look for bounds on Brun’s Constant. 1. Motivation and Background Twin prim...

Matrix Powers of Column-justified Pascal Triangles and Fibonacci Sequences

It is known that if Ln, respectively Rn9 are n x n matrices with the (/, j ) * entry the binomial coefficient (y~l)? respectively (^l)), then Ln = In (mod 2), respectively R„=In (mod 2), where In is the identity matrix of dimension n>\ (see, e.g., Problem PI 073 5 in the May 1999 issue of Arner. Math Monthly). The entries of Ln form a left-justified Pascal triangle and the entries of Rn result ...