Erdős asked: when does the base 3 expansion of a power of 2 omit the digit 2? His conjectured answer is that this only happens for 1, 4, and 256, but this conjecture is still open, and has proven to be very elusive. There underlies a deep relationship between the primes 2 and 3. Our attempt to understand this relationship has led to interesting connections among symbolic dynamical systems, graph theory, p-adic analysis, number theory, and fractal geometry. Despite the awesome variety of mathematics involved, linear algebra should be sufficient background knowledge for this talk. I report on joint work with Jeff Lagarias of the University of Michigan and Artem Bolshakov of the University of Texas at Dallas.