65th Annual STS (2005-2006)
Finalists
Nicholas Michael Wage

WISCONSIN
Nicholas Michael Wage, 17, of Appleton, studied generalized Paley graphs, an important class of graphs, for his Intel Science Talent Search project in mathematics. Given a prime p such that 4 divides p-1, we obtain a Paley graph by taking as vertices the integers (0, 1, ..., p-1), with an edge between x and y just in case x - y is a square modulo p. These, together with similarly defined graphs and directed graphs form the class called "generalized Paley." In the case above, when p - 1 is divisible by 4, Nick found the asymptotic limit, as p increases, for the number of complete subgraphs of a fixed size. He showed that this limit equaled that which Paul Erd”s (incorrectly) conjectured for all graphs. Nick also counted the number of three cycles for members of the larger family of generalized Paley graphs. His proofs used results from number theory, including Weil's deep theorem on the Riemann Hypothesis for finite fields. Nick, who attends Appleton East High School, earned 800s on his critical reading and math SAT scores. His paper is published in the journal Integers. Son of Drs. Michael Wage and Kathy Vogel, he plans to study math at Harvard or the University of Wisconsin.