Prerequisites are a solid background in high school algebra, geometry, and trigonometry, a healthy sense of curiosity about mathematical ideas, and a willingness to ask questions and work hard to answer them. Mathematical topics include: the fundamental theorem of arithmetic; elementary ring theory; unique factorization into irreducible elements, examples and counterexamples; probabilistic methods in the theory of prime numbers; the Riemann Hypothesis and why it matters. Strategies of mathematical investigation, including experimentation and observation, and the use of language as tool for investigation, will be a central theme of this seminar.