Prime Numbers in Generalized Pascal Triangles

Abstract

In this paper we generalize the Pascal triangle and examine the connectionsamong the generalized triangles and powering integers respectivelypolynomials. We emphasize connections with the binomial and multinomialtheorems, and we present some computational results in topics of primenumbers and prime factorization. Some prime numbers (candidates) connectedto the generalized Pascal triangle are presented. We describe a very efficientprimality proving algorithm called ECPP (Elliptic Curve Primality Proving)and for one of the candidates an exact primality proof is given. Since thisprime number has more than 1000 digits, it is a so-called “titanic prime”.