Primes and Orbits

We review various classical problems which are concerned with looking for primes or numbers with few prime factors. We then put these in a geometric and group theoretic context of values at polynomials on orbits of an action of affine space which preserves Zⁿ. The development of the combinatorial sieve in this context presents a number of novel features one of which is that certain graphs associated with these orbits be "expanders". We will give applications to classical problems such as the divisibilty of the areas of pythagorean triangles.