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  1. In this paper, we investigate the existence of Sierpi´nski numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer r, there exist infinitely many Sierpi´nski numbers and Riesel numbers of the form kCr. Let S(x) be the number of positive integers r satisfying 1 ≤ r ≤ x for which kCr is a Sierpi´nski number for infinitely many k. We further show that the value S(x)/x gets arbitrarily close to 1 as x tends to infinity. Generalizations to base a-Sierpi´nski numbers and base a-Riesel numbers are also considered. In particular, we prove that there exist infinitely many positive integers r such that kCr is simultaneously a base a-Sierpi´nski and base a-Riesel number for infinitely many k.
  2. Node-Kayles is an impartial game played on a simple graph. The Sprague-Grundy theorem states that every impartial game is associated with a nonnegative integer value called a Nimber. This paper studies the Nimber sequences of various families of graphs, including 3-paths, lattice graphs, prism graphs, chained cliques, linked cliques, linked cycles, linked diamonds, hypercubes, and generalized Petersen graphs. For most of these families, we determine an explicit formula or a recursion on their Nimber sequences.