Partial semigroups and primality indicators in the fractal generation of binomial coefficients to a prime square modulus

• Autorzy: Doan D. , Kivunge B. , Smith J. , Sykes T. , Teplitskiy M.
• Języki publikacji: EN

Abstrakty

EN

This paper, resulting from two summer programs of Research Experience for Undergraduates, examines the congruence classes of binomial coefficients to a prime square modulus as given by a fractal generation process for lattice path counts. The process depends on the isomorphism of partial semigroup structures associated with each iteration. We also consider integrality properties of certain critical coefficients that arise in the generation process. Generalizing the application of these coefficients to arbitrary arguments, instead of just to the prime arguments appearing in their original function, it transpires that integrality of the coefficients is indicative of the primality of the argument.

Słowa kluczowe

EN

binomial coefficients; Pascal's triangle; partial algebra

PL

współczynniki dwumianowe; trójkąt Pascala; algebra częściowa

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2009
• Tom: Vol. 42, nr 1
• Strony: 23--39
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Doan D.

autor

Kivunge B.

autor

Smith J.

autor

Sykes T.

autor

Teplitskiy M.

• Department of Mathematics Iowa State University Ames, Iowa 50011-2064, USA,

Bibliografia

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• [3] F. v. Haeseler, H.-O. Peitgen, G. Skorder, Pascal’s triangle, dynamical systems and attractors, Ergodic Th. and Dynamical Systems, 12 (1992), 479-486.
• [4] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1968.
• [5] E. E. Kummer, Über Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math. 44 (1852), 93-146.
• [6] H.-O. Peitgen, H. Jürgens, D. Saupe, Fractals for the Classroom, Part II, Springer, New York, NY, 1992.
• [7] A. B. Romanowska, J. D. H. Smith, Modes, World Scientific, River Edge, NJ, 2002.
• [8] M. Sved, The geometry of the binomial array modulo p2 and p3, Discrete Math. 92 (1991), 395-416.
• [9] M. Sved, J. Pitman, Divisibility of binomials by prime powers, Ars Combinatoria 26 (1988), 197-222.