# Sifting the Primes

**Abstract:**
These are the slides from a colloquium I presented at the University
of Hawai'i on 18 March 2005. I begin by discussing the famous sieve
of Eratosthenes, then demonstrate the power of sieve methods in
tackling such outstanding problems as the Goldbach conjecture. In
particular, I examine the weighted sieve of Chen Jing-Run, which he
applied in 1974 to demonstrate that every sufficiently large even
number N can be written in the form N = p + P_{2},
where P_{2} denotes an integer with at most 2 prime factors.

I go on to discuss some of my recent work in which I proved a similar 'almost-primes' result concerning the values taken by pairs of irreducible binary quadratic forms. In the same sense that Chen's result is an approximation to the Goldbach conjecture, my own result is an approximation to Schinzel's celebrated hypothesis H.

**Addendum: **Since I presented the talk, I have been
made aware of a paper of Diamond and Halberstam which extends the work
of Halberstam and Richert to give a P_{7} result to the pairs
of quadratics problem. Utilising their work, I reduced the number of
prime factors in my problem from 6 to 5.