Department of

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Thursday, November 12, 2015

**Abstract:** Let $A$ denote the set of Newman polynomials, $\sum c_i x^i, c_i \in \{0,1\}$. Which integers $m$ may be written as $m = \frac{p(3)}{q(3)}$, where $p, q \in A$? There are two complementary approaches: the algebraic and the combinatorial. In the first, one can show that 4 arises only when $p(x) = (1+x)q(x)$ and if 22 arises, then $q$ cannot divide $p$. In the second, all possible $q$'s are encoded by certain labeled closed walks in an uncomplicated digraph; in this way it can be shown that 529 and 592 can never occur. No particular prerequisites for this seminar.