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Colloquium:Density Properties of Pascal's Triangle Type Arrays
4:10 p.m. Neill Hall 5W
Abstract: Pascal's triangle modulo a prime exhibits striking properties, including patterns reminiscent of the Serpinski gasket. Such fractal patterns can be exploited to show that for any prime p, almost all entries in Pascal's triangle are divisible by p. Indeed, a published extension of this result declares that for almost all choices of nonnegative integers d, n, and k, the coefficient of xk in (1 + x + . . . + xd)n is divisible by p. In this talk, we use results from stochastic processes to extend this result for any polynomial P(x) with integer coefficients, i.e., for almost all choices of k and n, the coefficient of xk in the expansion of Q(x)n is divisible by p. Also, we show that this is true for reciprocals of such polynomials as well. Finally, we show that this result does not extend to all rational functions with integer coefficients by exhibiting a counterexample.