Numbers, polynomials and randomness

The basic tension between addition and multiplication has kept humans interested in integers for millennia. It is hardly surprising that prime numbers, the so-called atoms of arithmetic, lie at the heart of several schemes to detect or communicate withdistant civilisations. There are many interesting sequences of integers that reflect the behaviour of primes in one way or another. In this colloquium I'll talk about some special sequences drawn from the set {-1,+1} and recent attempts to identify randomness in them. I'll also discuss how similar ideas yield insight into basic questions about the existence of integer solutions to random polynomial equations.  This talk will be accessible to non-mathematicians.