# A Pascal-Type Triangle Characterizing Twin Primes

Year of Award: 2006

Award: Lester R. Ford

Publication Information: The American Mathematical Monthly, vol. 112, (2005), pp. 673-681

Summary: (From MAA Online) It is a well-known property of Pascal's triangle that the entries of the $$k$$th row, without the initial and final entries $$1$$, are all divisible by $$k$$ if and only if $$k$$ is prime. This paper presents a triangular array, analogous to Pascal's, that characterizes twin prime pairs in a similar fashion. The proof involves generating function techniques. Connections with orthogonal polynomials, in particular Chebyshev and ultraspherical polynomials, are also discussed.

About the Authors: [from The American Mathematical Monthly, (2005)]

Karl Dilcher received his undergraduate education at the Technische Universität Clausthal in Germany. He then did his graduate studies at Queen’s University in Kingston, Ontario, and finished his Ph.D. there in 1983 under the supervision of Paulo Ribenboim. He is currently a professor at Dalhousie University in Halifax, Nova Scotia, Canada, where he first arrived in 1984 as a postdoctoral fellow. His research interests include classical analysis, special functions, and elementary and computational number theory.

Kenneth B. Stolarsky received his undergraduate education at Caltech and obtained a Ph.D.from the University of Wisconsin (Madison) in 1968 under the supervision of Marvin I. Knopp. He did one year of postdoctoral work at the Institute for Advanced Study (Princeton) and has been (aside from sabbaticals at the University of Colorado (Boulder) and Yale University (New Haven)) at the University of Illinois (Urbana- Champaign) since then, where he is presently professor of mathematics. His research interests include number theory, extremal problems of geometry, and classical analysis.

Author (old format):
Karl Dilcher, Kenneth B. Stolarsky
Author(s):
Karl Dilcher and Kenneth B. Stolarsky
Flag for Digital Object Identifier:
Publication Date:
Wednesday, October 22, 2008
Publish Page:
Summary:

(From MAA Online) It is a well-known property of Pascal's triangle that the entries of the kth row, without the initial and final entries 1, are all divisible by k if and only if k is prime. This paper presents a triangular array, analogous to Pascal's, that characterizes twin prime pairs in a similar fashion. The proof involves generating function techniques. Connections with orthogonal polynomials, in particular Chebyshev and ultraspherical polynomials, are also discussed.