Matthew Michael Wage, 17, of Appleton, submitted an Intel Science Talent Search mathematics project that extended earlier results on arithmetic functions. The starting point for Matt’s project in number theory is Lehmer’s Conjecture, still open, that an arithmetic function defined by Ramanujan, the tau-function, is nonzero at each natural number n. Murty, Murty and Shorey showed that tau takes on any given value only finitely often. Matt extends this result to a wider class of arithmetic functions, sometimes at the cost of adding restrictions to the choice of n. Matt attends Appleton High School East where he is active in varsity football, varsity tennis and the ping pong club. Matt has won regional competitions in math, and his volunteer efforts as a coach helped the school’s math team earn the top rank in the state. He also enjoys playing chess, bridge and guitar. Matt’s quest for understanding the world around him has fueled his passion to learn everything from ideal gas laws to the propaganda genius of Genghis Khan. The son of Michael Wage and Kathy Vogel, Matt plans to study mathematics and medicine and pursue a career as a physician or mathematician.